{"id":1297,"date":"2026-01-26T18:01:18","date_gmt":"2026-01-26T08:01:18","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1297"},"modified":"2026-01-27T12:16:20","modified_gmt":"2026-01-27T02:16:20","slug":"year12-math-1-2-26-investing-and-networking%ef%bc%88topic2","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1297","title":{"rendered":"Year12-math-1-4-1 Investing and Networking\uff08Topic2)"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">In practical financial applications and graph theory. It covers compound interest, loans, annuities, and perpetuity calculations (Topic 1) , alongside network concepts such as graphs, paths, trails, and,Eulerian\/Hamiltonian graphs for modeling decision-making (Topic 2).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Topics in Unit 4: General Mathematics<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Topic 1: Loans, Investments and Annuities<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Compound Interest:<\/strong> Calculations, using recurrence relations, and applying the compound interest formula.<\/li>\n\n\n\n<li><strong>Financial Applications:<\/strong> Reducing-balance loans, annuities, and perpetuities.<\/li>\n\n\n\n<li><strong>Rates:<\/strong> Understanding effective annual rates of interest.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Topic 2: Graphs and Networks<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Fundamentals:<\/strong> Definitions of vertices, edges, directed graphs (digraphs), and subgraphs.<\/li>\n\n\n\n<li><strong>Matrix Representation:<\/strong> Creating and using adjacency matrices.<\/li>\n\n\n\n<li><strong>Graph Types:<\/strong> Planar graphs and Euler&#8217;s formula (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>V<\/mi><mo>\u2212<\/mo><mi>E<\/mi><mo>+<\/mo><mi>F<\/mi><mo>=<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"text\/plain\">cap V minus cap E plus cap F equals 2<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Paths and Circuits:<\/strong> Exploring walks, trails, paths, Eulerian graphs, and Hamiltonian graphs.&nbsp;<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Topic<\/strong> <strong>3. Key Mathematical Tools &amp; Skills<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Matrices:<\/strong>&nbsp;Used to represent graphs and network data.<\/li>\n\n\n\n<li><strong>Sequences and Series:<\/strong>&nbsp;Used to model the growth of investments and decay of loans.<\/li>\n\n\n\n<li><strong>Technology:<\/strong>&nbsp;Using spreadsheets and graphics calculators to perform calculations for annuities and network algorithms.&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">This unit prepares students for real-world financial management and using networks to solve optimization problems.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In the Queensland (QLD) Year 12 General Mathematics syllabus,&nbsp;<strong>Graphs and Networks<\/strong>&nbsp;(primarily found in Unit 4) and&nbsp;<strong>Loans, Investments and Annuities<\/strong>&nbsp;(often referred to alongside investing and financial networking) are key components of the curriculum.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is a breakdown of the key topics, focusing on the &#8220;Graphs and Networks&#8221; and &#8220;Investing&#8221; aspects:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Topic<\/strong> <strong>2. Graphs and Networks <\/strong>(Unit 4: Networks and Decision Mathematics)&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This topic focuses on using graph theory to solve practical, real-world problems involving connections.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Basic Concepts:<\/strong>&nbsp;Defining vertices (nodes) and edges (lines\/connections).<\/li>\n\n\n\n<li><strong>Types of Graphs:<\/strong>&nbsp;Understanding simple graphs, digraphs (directed graphs), subgraphs, and planar graphs (graphs that can be drawn without crossing lines).<\/li>\n\n\n\n<li><strong>Network Representation:<\/strong>&nbsp;Using adjacency matrices to represent relationships in a network.<\/li>\n\n\n\n<li><strong>Paths and Circuits:<\/strong>&nbsp;Identifying walks, trails, paths, circuits, and cycles (Eulerian trails\/circuits and Hamiltonian paths\/cycles).<\/li>\n\n\n\n<li><strong>Trees and Connectivity:<\/strong>&nbsp;Understanding spanning trees and minimum spanning trees (MST).<\/li>\n\n\n\n<li><strong>Algorithms:<\/strong>&nbsp;Solving problems using Kruskal\u2019s or Prim\u2019s algorithm (for minimum spanning trees), Dijkstra\u2019s algorithm (for shortest paths), and the Hungarian algorithm (for assignment problems).<\/li>\n\n\n\n<li><strong>Flows and Scheduling:<\/strong>&nbsp;Determining maximum flow and minimum cut in networks, as well as critical paths (scheduling).&nbsp;<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Topic 3. Key Mathematical Tools &amp; Skills<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Matrices:<\/strong>&nbsp;Used to represent graphs and network data.<\/li>\n\n\n\n<li><strong>Sequences and Series:<\/strong>&nbsp;Used to model the growth of investments and decay of loans.<\/li>\n\n\n\n<li><strong>Technology:<\/strong>&nbsp;Using spreadsheets and graphics calculators to perform calculations for annuities and network algorithms.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">These topics are designed to model real-world scenarios such as communication networks, transportation routes, project <\/p>\n\n\n\n<p class=\"wp-block-paragraph\">***************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 12 General Mathematics (Unit 4: Networks and Matrices\/Financial Math), graph theory is used to model, analyze, and optimize systems involving relationships. Here are the definitions of vertices, edges, directed graphs (digraphs), and subgraphs.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>1. Vertices (Nodes)&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> A vertex (plural: <em>vertices<\/em>) is a point, dot, or node that represents a specific object, location, or entity in a network.<\/li>\n\n\n\n<li><strong>Application:<\/strong> In a transportation network, a vertex could represent a city or an intersection. In a project network (CPA), it represents a milestone or the completion of an activity.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>2. Edges (Arcs\/Links)&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> An edge is a line segment that connects two vertices. It represents a relationship, connection, or path between the objects.<\/li>\n\n\n\n<li><strong>Types:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Undirected Edge:<\/strong> A line with no arrow, representing a two-way connection (e.g., a road you can drive on in both directions).<\/li>\n\n\n\n<li><strong>Directed Edge (Arc):<\/strong> A line with an arrow, indicating a one-way relationship.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Special Cases:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Loop:<\/strong> An edge that starts and ends at the same vertex.<\/li>\n\n\n\n<li><strong>Multiple Edges:<\/strong> Two or more edges connecting the same pair of vertices.&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>3. Directed Graphs (Digraphs)&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> A directed graph (or digraph) is a graph in which every edge has a direction assigned to it, shown by an arrow.<\/li>\n\n\n\n<li><strong>Characteristics:<\/strong> The edges are ordered pairs (often called arcs), connecting a &#8220;tail&#8221; vertex (source) to a &#8220;head&#8221; vertex (destination).<\/li>\n\n\n\n<li><strong>Application:<\/strong> Used for modeling one-way streets, dependency in project management (where task A must be done before task B), or flow networks.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>4. Subgraphs&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Definition:<\/strong> A subgraph is a graph formed by taking a subset of the vertices and a subset of the edges from a larger, original graph.<\/li>\n\n\n\n<li><strong>Rules:<\/strong> For a graph <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>H<\/mi><annotation encoding=\"text\/plain\">cap H<\/annotation><\/semantics><\/math> to be a subgraph of <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>G<\/mi><annotation encoding=\"text\/plain\">cap G<\/annotation><\/semantics><\/math>:\n<ol class=\"wp-block-list\">\n<li>All vertices in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>H<\/mi><annotation encoding=\"text\/plain\">cap H<\/annotation><\/semantics><\/math> must be present in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>G<\/mi><annotation encoding=\"text\/plain\">cap G<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li>All edges in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>H<\/mi><annotation encoding=\"text\/plain\">cap H<\/annotation><\/semantics><\/math> must be present in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>G<\/mi><annotation encoding=\"text\/plain\">cap G<\/annotation><\/semantics><\/math>.<\/li>\n<\/ol>\n<\/li>\n\n\n\n<li><strong>Application:<\/strong> A minimum spanning tree (used in algorithms to find the shortest total connection distance) is a type of subgraph.&nbsp;<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\">Summary Table of Graph Components&nbsp;<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><th>Term&nbsp;<\/th><th>Definition<\/th><th>Representation<\/th><\/tr><tr><td><strong>Vertex<\/strong><\/td><td>Points representing objects\/locations<\/td><td>Nodes (dots)<\/td><\/tr><tr><td><strong>Edge<\/strong><\/td><td>Lines connecting vertices<\/td><td>Lines\/Arcs<\/td><\/tr><tr><td><strong>Digraph<\/strong><\/td><td>Graph with one-way connections<\/td><td>Arrows<\/td><\/tr><tr><td><strong>Subgraph<\/strong><\/td><td>A portion of a larger graph<\/td><td>Subset of nodes\/lines<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Network Representation<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 12 General Mathematics Unit 4 (Networks and Decision Mathematics), a&nbsp;<strong>Network Representation<\/strong>&nbsp;is&nbsp;<mark>a mathematical model used to visualize and analyze systems of interconnected objects<\/mark>. It translates real-world scenarios\u2014such as transportation routes, social networks, or project tasks\u2014into a structured graph consisting of vertices (nodes) and edges (links).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is a breakdown of network representations as taught in this unit:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">1. Fundamental Components<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertex (Node):<\/strong>&nbsp;A dot or circle representing an object, location, or person.<\/li>\n\n\n\n<li><strong>Edge (Link\/Arc):<\/strong>&nbsp;A line connecting two vertices, representing a relationship (e.g., a road, connection, or flow).<\/li>\n\n\n\n<li><strong>Weighted Edge:<\/strong>&nbsp;An edge with a number indicating cost, distance, or time.<\/li>\n\n\n\n<li><strong>Directed Edge (Arc):<\/strong>&nbsp;An arrow representing a one-way relationship.<\/li>\n\n\n\n<li><strong>Undirected Edge:<\/strong>&nbsp;A line representing a two-way relationship.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">2. Common Types of Network Representations<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Graph\/Diagram:<\/strong>&nbsp;A visual drawing of vertices and edges, such as a map of cities (vertices) connected by roads (edges).<\/li>\n\n\n\n<li><strong>Weighted Graph:<\/strong>&nbsp;Used for finding the shortest path (e.g., travel times between towns).<\/li>\n\n\n\n<li><strong>Adjacency Matrix:<\/strong>&nbsp;A square matrix (table) that represents the number of connections between vertices. It uses 0s and 1s (or higher numbers) to show connections, with rows and columns representing vertices.<\/li>\n\n\n\n<li><strong>Digraph (Directed Graph):<\/strong>&nbsp;Used for one-way systems, such as a tournament result table.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">3. Key Applications in Unit 4<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Planar Graphs:<\/strong>&nbsp;Networks that can be drawn without edges crossing.<\/li>\n\n\n\n<li><strong>Trees and Spanning Trees:<\/strong>&nbsp;Subgraphs that connect all vertices with no cycles.<\/li>\n\n\n\n<li><strong>Flow Networks:<\/strong>&nbsp;Determining maximum flow (capacity) from a source to a sink.<\/li>\n\n\n\n<li><strong>Critical Path Analysis:<\/strong>&nbsp;Using networks to find the longest path (minimum time) to complete a project.<\/li>\n\n\n\n<li><strong>Hamiltonian Cycles\/Path:<\/strong>&nbsp;Finding paths that visit each vertex exactly once, often for &#8220;traveling salesperson&#8221; problems.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Example of Representation<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">A table showing travel times between five islands can be represented as a&nbsp;<strong>weighted graph<\/strong>&nbsp;(visual) or an&nbsp;<strong>adjacency matrix<\/strong>&nbsp;(tabular) to calculate the quickest route.&nbsp;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Paths and Circuits<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 12 General Mathematics (Unit 4: Networks and Decision Mathematics),&nbsp;<strong>Paths and Circuits<\/strong>&nbsp;are&nbsp;<mark>fundamental concepts in graph theory used to analyze, navigate, and solve optimization problems in networks<\/mark>. They describe specific types of routes (walks) within a network of vertices and edges.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Here is a breakdown of the key terminology and concepts:<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key Definitions (Paths, Circuits, and Associated Terms)<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Walk:<\/strong>&nbsp;A sequence of edges and vertices connecting points in a network; vertices and edges can be repeated.<\/li>\n\n\n\n<li><strong>Trail:<\/strong>&nbsp;A walk where&nbsp;<strong>no edges are repeated<\/strong>, but vertices can be repeated.<\/li>\n\n\n\n<li><strong>Path:<\/strong>&nbsp;A walk where&nbsp;<strong>no edges and no vertices are repeated<\/strong>&nbsp;(except possibly the start\/end in a cycle).<\/li>\n\n\n\n<li><strong>Circuit:<\/strong>&nbsp;A&nbsp;<strong>closed trail<\/strong>&nbsp;(starts and ends at the same vertex) where no edges are repeated.<\/li>\n\n\n\n<li><strong>Cycle:<\/strong>&nbsp;A&nbsp;<strong>closed path<\/strong>&nbsp;(starts and ends at the same vertex) where no vertices or edges are repeated.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Core Concepts in Unit 4<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Eulerian Trail:<\/strong>&nbsp;A trail that uses&nbsp;<strong>every edge<\/strong>&nbsp;in a graph exactly once. It starts and ends at different vertices (requires exactly two odd-degree vertices).<\/li>\n\n\n\n<li><strong>Eulerian Circuit:<\/strong>&nbsp;A circuit that uses&nbsp;<strong>every edge<\/strong>&nbsp;in a graph exactly once, starting and ending at the same vertex. This only occurs if&nbsp;<strong>every vertex<\/strong>&nbsp;in the graph has an even degree.<\/li>\n\n\n\n<li><strong>Hamiltonian Path\/Cycle:<\/strong>&nbsp;A path or cycle that visits&nbsp;<strong>every vertex<\/strong>&nbsp;exactly once.<\/li>\n\n\n\n<li><strong>Degree of a Vertex:<\/strong>&nbsp;The number of edges connected to a vertex.<\/li>\n\n\n\n<li><strong>Connected Graph:<\/strong>&nbsp;A graph where a path exists between all pairs of vertices.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Applications of Paths and Circuits<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">These concepts are used to solve practical &#8220;travelling&#8221; problems, including:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Eulerian Problems:<\/strong>&nbsp;Finding a route that covers every road\/street exactly once (e.g., mail delivery, street sweeping).<\/li>\n\n\n\n<li><strong>Hamiltonian Problems:<\/strong>&nbsp;Finding the shortest path visiting every city once (e.g., traveling salesman problem).<\/li>\n\n\n\n<li><strong>Minimum Connector Problems:<\/strong>&nbsp;Using Kruskal&#8217;s or Prim&#8217;s algorithm to connect all vertices with minimum edge weight.<\/li>\n\n\n\n<li><strong>Critical Path Analysis:<\/strong>&nbsp;Identifying the longest path in an activity network to determine the minimum time to complete a project.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Key Distinctions<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Path vs. Trail:<\/strong>&nbsp;Paths cannot repeat vertices; trails cannot repeat edges.<\/li>\n\n\n\n<li><strong>Circuit vs. Cycle:<\/strong>&nbsp;Circuits allow vertex repetition; cycles do not (except start\/end).<\/li>\n\n\n\n<li><strong>Eulerian vs. Hamiltonian:<\/strong>&nbsp;Eulerian deals with using every&nbsp;<strong>edge<\/strong>; Hamiltonian deals with visiting every&nbsp;<strong>vertex<\/strong>.&nbsp;<\/li>\n<\/ul>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Adjacency Matrix<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 12 General Mathematics (Unit 4: Networks and Decision Mathematics), an <strong>Adjacency Matrix<\/strong> is a square (<math data-latex=\"n x n\"><semantics><mrow><mi>n<\/mi><mi>x<\/mi><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n x n<\/annotation><\/semantics><\/math>) matrix used to represent a graph or network numerically. It records the number of direct connections (edges) between vertices, with rows representing the &#8220;starting&#8221; vertex and columns representing the &#8220;ending&#8221; vertex.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key Features and Definitions&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Structure:<\/strong> For a graph with <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> vertices, the adjacency matrix is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"text\/plain\">n cross n<\/annotation><\/semantics><\/math>. The matrix is arranged so that rows and columns correspond to the same set of vertices in the same order.<\/li>\n\n\n\n<li><strong>Values:<\/strong> An element <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msub><mi>A<\/mi><mrow><mi>i<\/mi><mi>j<\/mi><\/mrow><\/msub><annotation encoding=\"text\/plain\">cap A sub i j end-sub<\/annotation><\/semantics><\/math> indicates the number of direct edges from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math> to vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>j<\/mi><annotation encoding=\"text\/plain\">j<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><strong>Simple Graphs:<\/strong> For simple graphs (no loops, no multiple edges), the matrix contains only 0s and 1s, with 0s along the main diagonal.<\/li>\n\n\n\n<li><strong>Undirected Graphs:<\/strong> The matrix is <strong>symmetric<\/strong> across the main diagonal (i.e., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>A<\/mi><mrow><mi>i<\/mi><mi>j<\/mi><\/mrow><\/msub><mo>=<\/mo><msub><mi>A<\/mi><mrow><mi>j<\/mi><mi>i<\/mi><\/mrow><\/msub><\/mrow><annotation encoding=\"text\/plain\">cap A sub i j end-sub equals cap A sub j i end-sub<\/annotation><\/semantics><\/math>) because edges are bidirectional.<\/li>\n\n\n\n<li><strong>Directed Graphs (Digraphs):<\/strong> The matrix is generally <strong>not symmetric<\/strong> because edges have a specific direction (from <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>j<\/mi><annotation encoding=\"text\/plain\">j<\/annotation><\/semantics><\/math> does not mean <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>j<\/mi><annotation encoding=\"text\/plain\">j<\/annotation><\/semantics><\/math> to <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Loops:<\/strong> If a loop exists at a vertex, it is indicated by a 1 (or sometimes 2, depending on convention) on the main diagonal. However, in standard network representation, a loop counts as one edge.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Applications in Unit 4&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Determining Degree:<\/strong> The sum of the elements in a row (or column) of an adjacency matrix gives the degree of the corresponding vertex. Note: If loops exist, they are often counted twice when calculating the degree.<\/li>\n\n\n\n<li><strong>Counting Paths (<\/strong><math data-latex=\"M^n\"><semantics><msup><mi>M<\/mi><mi>n<\/mi><\/msup><annotation encoding=\"application\/x-tex\">M^n<\/annotation><\/semantics><\/math><strong>):<\/strong> By calculating matrix powers (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>A<\/mi><mn>2<\/mn><\/msup><mo>,<\/mo><msup><mi>A<\/mi><mn>3<\/mn><\/msup><\/mrow><annotation encoding=\"text\/plain\">cap A squared comma cap A cubed<\/annotation><\/semantics><\/math>, etc.), you can determine the number of paths of a specific length between vertices.\n<ul class=\"wp-block-list\">\n<li><math data-latex=\"\ud835\udc34^2\"><semantics><msup><mi>\ud835\udc34<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">\ud835\udc34^2<\/annotation><\/semantics><\/math><strong>:<\/strong> The element <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>i<\/mi><mo>,<\/mo><mi>j<\/mi><\/mrow><annotation encoding=\"text\/plain\">i comma j<\/annotation><\/semantics><\/math> in <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><msup><mi>A<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"text\/plain\">cap A squared<\/annotation><\/semantics><\/math> shows the number of 2-stage paths (paths with 2 edges) from vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>i<\/mi><annotation encoding=\"text\/plain\">i<\/annotation><\/semantics><\/math> to vertex <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>j<\/mi><annotation encoding=\"text\/plain\">j<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Identifying Connectedness:<\/strong> The adjacency matrix can be used to determine if a graph is connected.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Example: Undirected Graph&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">For an undirected graph with vertices {A, B, C} where A connects to B, and B connects to C:&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mtable><mtr><mtd><mrow><\/mrow><\/mtd><mtd><mi>A<\/mi><\/mtd><mtd><mi>B<\/mi><\/mtd><mtd><mi>C<\/mi><\/mtd><\/mtr><mtr><mtd><mi>A<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd><mi>B<\/mi><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd><mi>C<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><\/mtr><\/mtable><annotation encoding=\"text\/plain\">4 lines; Line 1: cap A cap B cap C; Line 2: cap A 0 1 0; Line 3: cap B 1 0 1; Line 4: cap C 0 1 0 end-lines;<\/annotation><\/semantics><\/math><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><em>Here, row A\/Col B is 1, and Row B\/Col A is 1 (symmetric).<\/em>&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Example: Directed Graph&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">If the edge goes from A to B only (A<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo>\u2192<\/mo><annotation encoding=\"text\/plain\">right arrow<\/annotation><\/semantics><\/math> B), then <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>A<\/mi><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">cap A sub cap A cap B end-sub equals 1<\/annotation><\/semantics><\/math> but <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msub><mi>A<\/mi><mrow><mi>B<\/mi><mi>A<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">cap A sub cap B cap A end-sub equals 0<\/annotation><\/semantics><\/math>.&nbsp;<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Digraph (Directed Graph)<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In QLD Year 12 General Mathematics (Unit 4: Networks and Decision Mathematics), a <strong>Digraph<\/strong> (short for <strong>Directed Graph<\/strong>) is <mark>a type of graph where the edges have a specific direction<\/mark>. Instead of representing a simple two-way connection (like a standard graph), the edges in a digraph are represented by arrows, indicating a one-way relationship.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Key Components and Characteristics&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Vertices (Nodes):<\/strong> The dots or points, representing objects or locations.<\/li>\n\n\n\n<li><strong>Arcs (Directed Edges):<\/strong> The arrows connecting the vertices. The direction is critical, representing, for example, &#8220;sharks eat fish&#8221; (one-way) rather than &#8220;sharks and fish share a border&#8221;.<\/li>\n\n\n\n<li><strong>Ordered Pairs:<\/strong> Each arc <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>(<\/mo><mi>u<\/mi><mo>,<\/mo><mi>v<\/mi><mo>)<\/mo><\/mrow><annotation encoding=\"text\/plain\">open paren u comma v close paren<\/annotation><\/semantics><\/math> is an ordered pair of vertices, pointing from the source (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>u<\/mi><annotation encoding=\"text\/plain\">u<\/annotation><\/semantics><\/math>) to the target (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>v<\/mi><annotation encoding=\"text\/plain\">v<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Simple Digraph:<\/strong> A digraph with no loops (edges starting and ending at the same vertex) or multiple edges between the same two vertices.<\/li>\n\n\n\n<li><strong>Weighted Digraph:<\/strong> A directed graph where a number (weight) is assigned to each arrow, representing metrics like distance, time, or cost.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Concepts in Unit 4&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Indegree:<\/strong> The number of arrows pointing <em>to<\/em> a vertex.<\/li>\n\n\n\n<li><strong>Outdegree:<\/strong> The number of arrows pointing <em>away<\/em> from a vertex.<\/li>\n\n\n\n<li><strong>Connectedness:<\/strong> A digraph is &#8220;connected&#8221; (or weakly connected) if the underlying graph is connected when all arrows are replaced by lines. It is &#8220;strongly connected&#8221; if there is a directed path from every vertex to every other vertex.<\/li>\n\n\n\n<li><strong>Representations:<\/strong> Digraphs are often represented using adjacency matrices, where entries represent the number of arcs from one vertex to another.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Examples and Applications&nbsp;<\/strong><\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>One-Way Systems:<\/strong> Road networks with one-way streets.<\/li>\n\n\n\n<li><strong>Tournament Graphs:<\/strong> Representing wins\/losses in a round-robin tournament.<\/li>\n\n\n\n<li><strong>Information\/Flow:<\/strong> Describing the flow of electricity, information, or materials in a network.&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">In summary, a digraph in QLD General Maths focuses on modeling systems where the direction of the relationship, flow, or path is crucial.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Topic 2:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"> \u30b0\u30e9\u30d5\u3068\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\uff08\u30e6\u30cb\u30c3\u30c84\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u610f\u601d\u6c7a\u5b9a\u6570\u5b66\uff09<br>\u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u30b0\u30e9\u30d5\u7406\u8ad6\u3092\u7528\u3044\u3066\u3001\u63a5\u7d9a\u3092\u542b\u3080\u5b9f\u4e16\u754c\u306e\u5b9f\u7528\u7684\u306a\u554f\u984c\u3092\u89e3\u304f\u3053\u3068\u306b\u7126\u70b9\u3092\u5f53\u3066\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u57fa\u672c\u6982\u5ff5<\/strong>\uff1a\u9802\u70b9\uff08\u30ce\u30fc\u30c9\uff09\u3068\u8fba\uff08\u7dda\uff0f\u63a5\u7d9a\uff09\u306e\u5b9a\u7fa9\u3002<br><strong>\u30b0\u30e9\u30d5\u306e\u7a2e\u985e<\/strong>\uff1a\u5358\u7d14\u30b0\u30e9\u30d5\u3001\u6709\u5411\u30b0\u30e9\u30d5\u3001\u90e8\u5206\u30b0\u30e9\u30d5\u3001\u5e73\u9762\u30b0\u30e9\u30d5\uff08\u7dda\u3092\u4ea4\u5dee\u3055\u305b\u305a\u306b\u63cf\u753b\u3067\u304d\u308b\u30b0\u30e9\u30d5\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<br><strong>\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe<\/strong>\uff1a\u96a3\u63a5\u884c\u5217\u3092\u7528\u3044\u3066\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u5185\u306e\u95a2\u4fc2\u3092\u8868\u73fe\u3059\u308b\u3002<br><strong>\u30d1\u30b9\u3068\u56de\u8def<\/strong>\uff1a\u30a6\u30a9\u30fc\u30af\u3001\u30c8\u30ec\u30a4\u30eb\u3001\u30d1\u30b9\u3001\u56de\u8def\u3001\u30b5\u30a4\u30af\u30eb\uff08\u30aa\u30a4\u30e9\u30fc\u30fb\u30c8\u30ec\u30a4\u30eb\uff0f\u56de\u8def\u3068\u30cf\u30df\u30eb\u30c8\u30f3\u30fb\u30d1\u30b9\uff0f\u30b5\u30a4\u30af\u30eb\uff09\u3092\u8b58\u5225\u3059\u308b\u3002<br><strong>\u6728\u3068\u9023\u7d50\u6027<\/strong>\uff1a\u5168\u57df\u6728\u3068\u6700\u5c0f\u5168\u57df\u6728\uff08MST\uff09\u3092\u7406\u89e3\u3059\u308b\u3002<br><strong>\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0<\/strong>\uff1a\u30af\u30e9\u30b9\u30ab\u30eb\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u307e\u305f\u306f\u30d7\u30ea\u30e0\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff08\u6700\u5c0f\u5168\u57df\u6728\uff09\u3001\u30c0\u30a4\u30af\u30b9\u30c8\u30e9\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff08\u6700\u77ed\u7d4c\u8def\uff09\u3001\u30cf\u30f3\u30ac\u30ea\u30a2\u30f3\u30fb\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\uff08\u5272\u308a\u5f53\u3066\u554f\u984c\uff09\u3092\u7528\u3044\u3066\u554f\u984c\u3092\u89e3\u304f\u3002<br><strong>\u30d5\u30ed\u30fc\u3068\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0<\/strong>\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u304a\u3051\u308b\u6700\u5927\u30d5\u30ed\u30fc\u3068\u6700\u5c0f\u30ab\u30c3\u30c8\u3001\u304a\u3088\u3073\u30af\u30ea\u30c6\u30a3\u30ab\u30eb\u30d1\u30b9\uff08\u30b9\u30b1\u30b8\u30e5\u30fc\u30ea\u30f3\u30b0\uff09\u3092\u6c7a\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Topic 3:<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u8981\u306a\u6570\u5b66\u7684\u30c4\u30fc\u30eb\u3068\u30b9\u30ad\u30eb<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u884c\u5217<\/strong>\uff1a\u30b0\u30e9\u30d5\u3084\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u30c7\u30fc\u30bf\u3092\u8868\u3059\u305f\u3081\u306b\u4f7f\u7528\u3059\u308b\u3002<br><strong>\u6570\u5217\u3068\u7d1a\u6570<\/strong>\uff1a\u6295\u8cc7\u306e\u5897\u52a0\u3068\u30ed\u30fc\u30f3\u306e\u6e1b\u5c11\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u308b\u3002<br><strong>\u30c6\u30af\u30ce\u30ed\u30b8\u30fc<\/strong>\uff1a\u30b9\u30d7\u30ec\u30c3\u30c9\u30b7\u30fc\u30c8\u3068\u30b0\u30e9\u30d5\u96fb\u5353\u3092\u7528\u3044\u3066\u3001\u5e74\u91d1\u3084\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306e\u8a08\u7b97\u3092\u884c\u3046\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u308c\u3089\u306e\u30c8\u30d4\u30c3\u30af\u306f\u3001\u901a\u4fe1\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3001\u8f38\u9001\u30eb\u30fc\u30c8\u3001\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u7ba1\u7406\uff08\u30b9\u30b1\u30b8\u30e5\u30fc\u30eb\u7ba1\u7406\uff09\u3001\u8ca1\u52d9\u8a08\u753b\u3068\u3044\u3063\u305f\u73fe\u5b9f\u4e16\u754c\u306e\u30b7\u30ca\u30ea\u30aa\u3092\u30e2\u30c7\u30eb\u5316\u3059\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c84\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u884c\u5217\uff0f\u91d1\u878d\u6570\u5b66\uff09\u3067\u306f\u3001\u30b0\u30e9\u30d5\u7406\u8ad6\u3092\u7528\u3044\u3066\u3001\u95a2\u4fc2\u6027\u3092\u542b\u3080\u30b7\u30b9\u30c6\u30e0\u306e\u30e2\u30c7\u30eb\u5316\u3001\u5206\u6790\u3001\u6700\u9069\u5316\u3092\u884c\u3044\u307e\u3059\u3002\u3053\u3053\u3067\u306f\u3001\u9802\u70b9\u3001\u8fba\u3001\u6709\u5411\u30b0\u30e9\u30d5\u3001\u90e8\u5206\u30b0\u30e9\u30d5\u306e\u5b9a\u7fa9\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u9802\u70b9\uff08\u30ce\u30fc\u30c9\uff09\u5b9a\u7fa9\uff1a\u9802\u70b9\uff08\u8907\u6570\u5f62\uff1a\u9802\u70b9\uff09\u306f\u3001\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u5185\u306e\u7279\u5b9a\u306e\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3001\u5834\u6240\u3001\u307e\u305f\u306f\u30a8\u30f3\u30c6\u30a3\u30c6\u30a3\u3092\u8868\u3059\u70b9\u3001\u30c9\u30c3\u30c8\u3001\u307e\u305f\u306f\u30ce\u30fc\u30c9\u3067\u3059\u3002\u5fdc\u7528\uff1a\u4ea4\u901a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3067\u306f\u3001\u9802\u70b9\u306f\u90fd\u5e02\u3084\u4ea4\u5dee\u70b9\u3092\u8868\u3059\u3053\u3068\u304c\u3042\u308a\u307e\u3059\u3002\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\uff08CPA\uff09\u3067\u306f\u3001\u30de\u30a4\u30eb\u30b9\u30c8\u30fc\u30f3\u307e\u305f\u306f\u30a2\u30af\u30c6\u30a3\u30d3\u30c6\u30a3\u306e\u5b8c\u4e86\u3092\u8868\u3057\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u8fba\uff08\u30a2\u30fc\u30af\uff0f\u30ea\u30f3\u30af\uff09\u5b9a\u7fa9\uff1a\u8fba\u306f\u30012\u3064\u306e\u9802\u70b9\u3092\u7d50\u3076\u7dda\u5206\u3067\u3059\u3002\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u9593\u306e\u95a2\u4fc2\u3001\u63a5\u7d9a\u3001\u307e\u305f\u306f\u30d1\u30b9\u3092\u8868\u3057\u307e\u3059\u3002\u7a2e\u985e\uff1a\u7121\u5411\u30a8\u30c3\u30b8\uff1a\u77e2\u5370\u306e\u306a\u3044\u7dda\u3067\u3001\u53cc\u65b9\u5411\u306e\u63a5\u7d9a\u3092\u8868\u3057\u307e\u3059\uff08\u4f8b\uff1a\u4e21\u65b9\u5411\u306b\u901a\u884c\u3067\u304d\u308b\u9053\u8def\uff09\u3002\u6709\u5411\u30a8\u30c3\u30b8\uff08\u30a2\u30fc\u30af\uff09\uff1a\u77e2\u5370\u306e\u3042\u308b\u7dda\u3067\u3001\u4e00\u65b9\u5411\u306e\u95a2\u4fc2\u3092\u793a\u3057\u307e\u3059\u3002\u7279\u6b8a\u306a\u30b1\u30fc\u30b9\uff1a\u30eb\u30fc\u30d7\uff1a\u540c\u3058\u9802\u70b9\u3067\u59cb\u307e\u308a\u3001\u540c\u3058\u9802\u70b9\u3067\u7d42\u308f\u308b\u30a8\u30c3\u30b8\u3002\u8907\u6570\u30a8\u30c3\u30b8\uff1a\u540c\u3058\u9802\u70b9\u306e\u30da\u30a2\u3092\u63a5\u7d9a\u3059\u308b2\u3064\u4ee5\u4e0a\u306e\u30a8\u30c3\u30b8\u3002<\/li>\n\n\n\n<li>\u6709\u5411\u30b0\u30e9\u30d5\uff08\u30c0\u30a4\u30b0\u30e9\u30d5\uff09\u5b9a\u7fa9\uff1a\u6709\u5411\u30b0\u30e9\u30d5\uff08\u307e\u305f\u306f\u30c0\u30a4\u30b0\u30e9\u30d5\uff09\u306f\u3001\u3059\u3079\u3066\u306e\u30a8\u30c3\u30b8\u306b\u65b9\u5411\u304c\u5272\u308a\u5f53\u3066\u3089\u308c\u3001\u77e2\u5370\u3067\u793a\u3055\u308c\u308b\u30b0\u30e9\u30d5\u3067\u3059\u3002\u7279\u5fb4\uff1a\u30a8\u30c3\u30b8\u306f\u9806\u5e8f\u4ed8\u304d\u30da\u30a2\uff08\u30a2\u30fc\u30af\u3068\u547c\u3070\u308c\u308b\u3053\u3068\u304c\u591a\u3044\uff09\u3067\u3001\u300c\u672b\u5c3e\u300d\u306e\u9802\u70b9\uff08\u30bd\u30fc\u30b9\uff09\u3068\u300c\u5148\u982d\u300d\u306e\u9802\u70b9\uff08\u30c7\u30b9\u30c6\u30a3\u30cd\u30fc\u30b7\u30e7\u30f3\uff09\u3092\u63a5\u7d9a\u3057\u307e\u3059\u3002\u7528\u9014\uff1a\u4e00\u65b9\u901a\u884c\u306e\u9053\u8def\u3001\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u7ba1\u7406\u306b\u304a\u3051\u308b\u4f9d\u5b58\u95a2\u4fc2\uff08\u30bf\u30b9\u30afA\u3092\u30bf\u30b9\u30afB\u306e\u524d\u306b\u5b9f\u884c\u3059\u308b\u5fc5\u8981\u304c\u3042\u308b\u5834\u5408\uff09\u3001\u307e\u305f\u306f\u30d5\u30ed\u30fc\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u30b5\u30d6\u30b0\u30e9\u30d5\u5b9a\u7fa9\uff1a\u30b5\u30d6\u30b0\u30e9\u30d5\u3068\u306f\u3001\u3088\u308a\u5927\u304d\u306a\u5143\u306e\u30b0\u30e9\u30d5\u304b\u3089\u9802\u70b9\u306e\u30b5\u30d6\u30bb\u30c3\u30c8\u3068\u30a8\u30c3\u30b8\u306e\u30b5\u30d6\u30bb\u30c3\u30c8\u3092\u53d6\u5f97\u3057\u3066\u5f62\u6210\u3055\u308c\u308b\u30b0\u30e9\u30d5\u3067\u3059\u3002\u30eb\u30fc\u30eb\uff1a\u30b0\u30e9\u30d5 (H) \u304c (G) \u306e\u30b5\u30d6\u30b0\u30e9\u30d5\u306b\u306a\u308b\u305f\u3081\u306b\u306f\u3001\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002(H) \u306e\u3059\u3079\u3066\u306e\u9802\u70b9\u304c (G) \u306b\u5b58\u5728\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002(H) \u306e\u3059\u3079\u3066\u306e\u30a8\u30c3\u30b8\u304c (G) \u306b\u5b58\u5728\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059\u3002\u5fdc\u7528\uff1a\u6700\u5c0f\u5168\u57df\u6728\uff08\u6700\u77ed\u306e\u63a5\u7d9a\u8ddd\u96e2\u3092\u6c42\u3081\u308b\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3067\u4f7f\u7528\u3055\u308c\u308b\uff09\u306f\u3001\u30b5\u30d6\u30b0\u30e9\u30d5\u306e\u4e00\u7a2e\u3067\u3059\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u30b0\u30e9\u30d5\u69cb\u6210\u8981\u7d20\u306e\u6982\u8981\u8868<\/p>\n\n\n\n<figure class=\"wp-block-table\"><table class=\"has-fixed-layout\"><tbody><tr><th>\u7528\u8a9e<\/th><th>\u5b9a\u7fa9<\/th><th>\u8868\u73fe<\/th><\/tr><tr><td><strong>\u9802\u70b9<\/strong><\/td><td>\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\/\u4f4d\u7f6e\u3092\u8868\u3059\u70b9<\/td><td>\u30ce\u30fc\u30c9\uff08\u70b9\uff09<\/td><\/tr><tr><td><strong>\u30a8\u30c3\u30b8<\/strong><\/td><td>\u9802\u70b9\u3092\u63a5\u7d9a\u3059\u308b\u7dda<\/td><td>\u7dda\/\u5f27<\/td><\/tr><tr><td><strong>\u6709\u5411\u30b0\u30e9\u30d5<\/strong><\/td><td>\u4e00\u65b9\u5411\u63a5\u7d9a\u3092\u6301\u3064\u30b0\u30e9\u30d5<\/td><td>\u77e2\u5370<\/td><\/tr><tr><td><strong>\u30b5\u30d6\u30b0\u30e9\u30d5<\/strong><\/td><td>\u3088\u308a\u5927\u304d\u306a\u30b0\u30e9\u30d5\u306e\u4e00\u90e8<\/td><td>\u30ce\u30fc\u30c9\/\u7dda\u306e\u30b5\u30d6\u30bb\u30c3\u30c8<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\u30e6\u30cb\u30c3\u30c84\uff08\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u610f\u601d\u6c7a\u5b9a\u6570\u5b66\uff09\u3067\u306f\u3001\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe\u3068\u306f\u3001\u76f8\u4e92\u63a5\u7d9a\u3055\u308c\u305f\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u306e\u30b7\u30b9\u30c6\u30e0\u3092\u8996\u899a\u5316\u3057\u5206\u6790\u3059\u308b\u305f\u3081\u306b\u7528\u3044\u3089\u308c\u308b\u6570\u5b66\u30e2\u30c7\u30eb\u3067\u3059\u3002\u3053\u308c\u306f\u3001\u4ea4\u901a\u30eb\u30fc\u30c8\u3001\u30bd\u30fc\u30b7\u30e3\u30eb\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3001\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u30bf\u30b9\u30af\u3068\u3044\u3063\u305f\u73fe\u5b9f\u4e16\u754c\u306e\u30b7\u30ca\u30ea\u30aa\u3092\u3001\u9802\u70b9\uff08\u30ce\u30fc\u30c9\uff09\u3068\u8fba\uff08\u30ea\u30f3\u30af\uff09\u304b\u3089\u306a\u308b\u69cb\u9020\u5316\u30b0\u30e9\u30d5\u306b\u5909\u63db\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30e6\u30cb\u30c3\u30c8\u3067\u5b66\u7fd2\u3059\u308b\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe\u306e\u5185\u8a33\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u57fa\u672c\u69cb\u6210\u8981\u7d20<br>\u9802\u70b9\uff08\u30ce\u30fc\u30c9\uff09\uff1a\u30aa\u30d6\u30b8\u30a7\u30af\u30c8\u3001\u5834\u6240\u3001\u307e\u305f\u306f\u4eba\u7269\u3092\u8868\u3059\u70b9\u307e\u305f\u306f\u5186\u3002<br>\u8fba\uff08\u30ea\u30f3\u30af\/\u30a2\u30fc\u30af\uff09\uff1a2\u3064\u306e\u9802\u70b9\u3092\u7d50\u3076\u7dda\u3067\u3001\u95a2\u4fc2\u6027\uff08\u4f8b\uff1a\u9053\u8def\u3001\u63a5\u7d9a\u3001\u30d5\u30ed\u30fc\uff09\u3092\u8868\u3057\u307e\u3059\u3002<br>\u91cd\u307f\u4ed8\u304d\u8fba\uff1a\u30b3\u30b9\u30c8\u3001\u8ddd\u96e2\u3001\u307e\u305f\u306f\u6642\u9593\u3092\u793a\u3059\u6570\u5024\u3092\u6301\u3064\u8fba\u3002<br>\u6709\u5411\u8fba\uff08\u30a2\u30fc\u30af\uff09\uff1a\u4e00\u65b9\u5411\u306e\u95a2\u4fc2\u3092\u8868\u3059\u77e2\u5370\u3002<br>\u7121\u5411\u8fba\uff1a\u53cc\u65b9\u5411\u306e\u95a2\u4fc2\u3092\u8868\u3059\u7dda\u3002<\/li>\n\n\n\n<li>\u4e00\u822c\u7684\u306a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe\u306e\u7a2e\u985e<br>\u30b0\u30e9\u30d5\/\u30c0\u30a4\u30a2\u30b0\u30e9\u30e0\uff1a\u9802\u70b9\u3068\u8fba\u3092\u8996\u899a\u7684\u306b\u8868\u3057\u305f\u3082\u306e\u3002\u4f8b\u3048\u3070\u3001\u90fd\u5e02\uff08\u9802\u70b9\uff09\u304c\u9053\u8def\uff08\u8fba\uff09\u3067\u7d50\u3070\u308c\u305f\u5730\u56f3\u306a\u3069\u3002<br>\u91cd\u307f\u4ed8\u304d\u30b0\u30e9\u30d5\uff1a\u6700\u77ed\u7d4c\u8def\uff08\u4f8b\uff1a\u90fd\u5e02\u9593\u306e\u79fb\u52d5\u6642\u9593\uff09\u3092\u6c42\u3081\u308b\u969b\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<br>\u96a3\u63a5\u884c\u5217\uff1a\u9802\u70b9\u9593\u306e\u63a5\u7d9a\u6570\u3092\u8868\u3059\u6b63\u65b9\u884c\u5217\uff08\u8868\uff09\u30020\u30681\uff08\u307e\u305f\u306f\u305d\u308c\u4ee5\u4e0a\u306e\u6570\uff09\u3067\u63a5\u7d9a\u3092\u8868\u3057\u3001\u884c\u3068\u5217\u3067\u9802\u70b9\u3092\u8868\u3057\u307e\u3059\u3002<br>\u6709\u5411\u30b0\u30e9\u30d5\uff08\u6709\u5411\u30b0\u30e9\u30d5\uff09\uff1a\u30c8\u30fc\u30ca\u30e1\u30f3\u30c8\u7d50\u679c\u8868\u306a\u3069\u3001\u4e00\u65b9\u5411\u306e\u30b7\u30b9\u30c6\u30e0\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/li>\n\n\n\n<li>\u30e6\u30cb\u30c3\u30c84\u3067\u306e\u4e3b\u306a\u5fdc\u7528<br>\u5e73\u9762\u30b0\u30e9\u30d5\uff1a\u8fba\u304c\u4ea4\u5dee\u3059\u308b\u3053\u3068\u306a\u304f\u63cf\u753b\u3067\u304d\u308b\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3002<br>\u6728\u3068\u5168\u57df\u6728\uff1a\u3059\u3079\u3066\u306e\u9802\u70b9\u3092\u9589\u8def\u306a\u3057\u3067\u63a5\u7d9a\u3059\u308b\u90e8\u5206\u30b0\u30e9\u30d5\u3002<br>\u30d5\u30ed\u30fc\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\uff1a\u30bd\u30fc\u30b9\u304b\u3089\u30b7\u30f3\u30af\u3078\u306e\u6700\u5927\u30d5\u30ed\u30fc\uff08\u5bb9\u91cf\uff09\u3092\u6c7a\u5b9a\u3059\u308b\u3002<br>\u30af\u30ea\u30c6\u30a3\u30ab\u30eb\u30d1\u30b9\u5206\u6790\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3092\u7528\u3044\u3066\u3001\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u3092\u5b8c\u4e86\u3059\u308b\u305f\u3081\u306e\u6700\u9577\u7d4c\u8def\uff08\u6700\u77ed\u6642\u9593\uff09\u3092\u898b\u3064\u3051\u307e\u3059\u3002<br>\u30cf\u30df\u30eb\u30c8\u30f3\u9589\u8def\uff0f\u30d1\u30b9\uff1a\u5404\u9802\u70b9\u30921\u56de\u3060\u3051\u8a2a\u308c\u308b\u7d4c\u8def\u3092\u898b\u3064\u3051\u307e\u3059\u3002\u3053\u308c\u306f\u3001\u300c\u5de1\u56de\u30bb\u30fc\u30eb\u30b9\u30de\u30f3\u554f\u984c\u300d\u306a\u3069\u3067\u3088\u304f\u7528\u3044\u3089\u308c\u307e\u3059\u3002<\/li>\n<\/ol>\n\n\n\n<p class=\"wp-block-paragraph\">\u8868\u73fe\u4f8b<br>5\u3064\u306e\u5cf6\u9593\u306e\u79fb\u52d5\u6642\u9593\u3092\u793a\u3059\u8868\u306f\u3001\u91cd\u307f\u4ed8\u304d\u30b0\u30e9\u30d5\uff08\u8996\u899a\u7684\uff09\u307e\u305f\u306f\u96a3\u63a5\u884c\u5217\uff08\u8868\u5f62\u5f0f\uff09\u3068\u3057\u3066\u8868\u73fe\u3057\u3001\u6700\u77ed\u7d4c\u8def\u3092\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30d1\u30b9\u3068\u56de\u8def<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c84\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u6c7a\u5b9a\u6570\u5b66\uff09\u306b\u304a\u3044\u3066\u3001\u30d1\u30b9\u3068\u56de\u8def\u306f\u30b0\u30e9\u30d5\u7406\u8ad6\u306b\u304a\u3051\u308b\u57fa\u672c\u6982\u5ff5\u3067\u3042\u308a\u3001\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u306b\u304a\u3051\u308b\u6700\u9069\u5316\u554f\u984c\u306e\u5206\u6790\u3001\u30ca\u30d3\u30b2\u30fc\u30b7\u30e7\u30f3\u3001\u304a\u3088\u3073\u89e3\u6c7a\u306b\u7528\u3044\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u3089\u306f\u3001\u9802\u70b9\u3068\u8fba\u306e\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u5185\u306b\u304a\u3051\u308b\u7279\u5b9a\u306e\u7a2e\u985e\u306e\u7d4c\u8def\uff08\u30a6\u30a9\u30fc\u30af\uff09\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u8981\u306a\u7528\u8a9e\u3068\u6982\u5ff5\u3092\u4ee5\u4e0b\u306b\u8aac\u660e\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u8981\u306a\u5b9a\u7fa9\uff08\u30d1\u30b9\u3001\u56de\u8def\u3001\u304a\u3088\u3073\u95a2\u9023\u7528\u8a9e\uff09<\/strong><br><strong>\u30a6\u30a9\u30fc\u30af<\/strong>\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u5185\u306e\u70b9\u3092\u7d50\u3076\u8fba\u3068\u9802\u70b9\u306e\u5217\u3002\u9802\u70b9\u3068\u8fba\u306f\u7e70\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<br><strong>\u30c8\u30ec\u30a4\u30eb<\/strong>\uff1a\u8fba\u306f\u7e70\u308a\u8fd4\u3055\u308c\u307e\u305b\u3093\u304c\u3001\u9802\u70b9\u306f\u7e70\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u30a6\u30a9\u30fc\u30af\u3002<br><strong>\u30d1\u30b9<\/strong>\uff1a\u8fba\u3068\u9802\u70b9\u304c\u7e70\u308a\u8fd4\u3055\u308c\u306a\u3044\u30a6\u30a9\u30fc\u30af\uff08\u5faa\u74b0\u306e\u958b\u59cb\u70b9\u3068\u7d42\u4e86\u70b9\u3092\u9664\u304f\uff09\u3002<br><strong>\u56de\u8def<\/strong>\uff1a\u8fba\u304c\u7e70\u308a\u8fd4\u3055\u308c\u306a\u3044\u9589\u3058\u305f\u9053\uff08\u958b\u59cb\u70b9\u3068\u7d42\u4e86\u70b9\u304c\u540c\u3058\u9802\u70b9\uff09\u3002<br><strong>\u30b5\u30a4\u30af\u30eb<\/strong>\uff1a\u9802\u70b9\u3082\u8fba\u3082\u7e70\u308a\u8fd4\u3055\u308c\u306a\u3044\u9589\u3058\u305f\u30d1\u30b9\uff08\u958b\u59cb\u70b9\u3068\u7d42\u4e86\u70b9\u304c\u540c\u3058\u9802\u70b9\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30e6\u30cb\u30c3\u30c84\u306e\u4e3b\u8981\u6982\u5ff5<\/strong><br><strong>\u30aa\u30a4\u30e9\u30fc\u30c8\u30ec\u30a4\u30eb<\/strong>\uff1a\u30b0\u30e9\u30d5\u5185\u306e\u3059\u3079\u3066\u306e\u8fba\u30921\u56de\u3060\u3051\u4f7f\u7528\u3059\u308b\u30c8\u30ec\u30a4\u30eb\u3002\u958b\u59cb\u70b9\u3068\u7d42\u4e86\u70b9\u304c\u7570\u306a\u308b\u9802\u70b9\u3067\u884c\u308f\u308c\u307e\u3059\uff08\u5947\u6570\u6b21\u6570\u306e\u9802\u70b9\u304c2\u3064\u5fc5\u8981\u3067\u3059\uff09\u3002<br><strong>\u30aa\u30a4\u30e9\u30fc\u56de\u8def<\/strong>\uff1a\u30b0\u30e9\u30d5\u5185\u306e\u3059\u3079\u3066\u306e\u8fba\u30921\u56de\u3060\u3051\u4f7f\u7528\u3057\u3001\u958b\u59cb\u70b9\u3068\u7d42\u4e86\u70b9\u304c\u540c\u3058\u9802\u70b9\u3067\u884c\u308f\u308c\u308b\u56de\u8def\u3002\u3053\u308c\u306f\u3001\u30b0\u30e9\u30d5\u5185\u306e\u3059\u3079\u3066\u306e\u9802\u70b9\u306e\u6b21\u6570\u304c\u5076\u6570\u3067\u3042\u308b\u5834\u5408\u306b\u306e\u307f\u767a\u751f\u3057\u307e\u3059\u3002<br><strong>\u30cf\u30df\u30eb\u30c8\u30f3\u30d1\u30b9\/\u30b5\u30a4\u30af\u30eb<\/strong>\uff1a\u3059\u3079\u3066\u306e\u9802\u70b9\u3092\u3061\u3087\u3046\u30691\u56de\u8a2a\u308c\u308b\u30d1\u30b9\u307e\u305f\u306f\u30b5\u30a4\u30af\u30eb\u3002<br><strong>\u9802\u70b9\u306e\u6b21\u6570<\/strong>\uff1a\u9802\u70b9\u306b\u63a5\u7d9a\u3055\u308c\u3066\u3044\u308b\u8fba\u306e\u6570\u3002<br><strong>\u9023\u7d50\u30b0\u30e9\u30d5<\/strong>\uff1a\u3059\u3079\u3066\u306e\u9802\u70b9\u30da\u30a2\u9593\u306b\u30d1\u30b9\u304c\u5b58\u5728\u3059\u308b\u30b0\u30e9\u30d5\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u30d1\u30b9\u3068\u56de\u8def\u306e\u5fdc\u7528<\/strong><br>\u3053\u308c\u3089\u306e\u6982\u5ff5\u306f\u3001\u6b21\u306e\u3088\u3046\u306a\u5b9f\u7528\u7684\u306a\u300c\u5de1\u56de\u300d\u554f\u984c\u3092\u89e3\u6c7a\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u307e\u3059\u3002<br><strong>\u30aa\u30a4\u30e9\u30fc\u554f\u984c<\/strong>\uff1a\u3059\u3079\u3066\u306e\u9053\u8def\/\u8857\u8def\u3092\u3061\u3087\u3046\u30691\u56de\u901a\u904e\u3059\u308b\u7d4c\u8def\u3092\u898b\u3064\u3051\u308b\uff08\u4f8b\uff1a\u90f5\u4fbf\u914d\u9054\u3001\u9053\u8def\u6e05\u6383\uff09\u3002<br><strong>\u30cf\u30df\u30eb\u30c8\u30f3\u554f\u984c<\/strong>\uff1a\u3059\u3079\u3066\u306e\u90fd\u5e02\u30921\u56de\u305a\u3064\u8a2a\u308c\u308b\u6700\u77ed\u7d4c\u8def\u3092\u898b\u3064\u3051\u308b\uff08\u4f8b\uff1a\u5de1\u56de\u30bb\u30fc\u30eb\u30b9\u30de\u30f3\u554f\u984c\uff09\u3002<br><strong>\u6700\u5c0f\u30b3\u30cd\u30af\u30bf\u554f\u984c<\/strong>\uff1a\u30af\u30e9\u30b9\u30ab\u30eb\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u307e\u305f\u306f\u30d7\u30ea\u30e0\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u3092\u7528\u3044\u3066\u3001\u3059\u3079\u3066\u306e\u9802\u70b9\u3092\u6700\u5c0f\u306e\u8fba\u306e\u91cd\u307f\u3067\u63a5\u7d9a\u3059\u308b\u3002<br><strong>\u30af\u30ea\u30c6\u30a3\u30ab\u30eb\u30d1\u30b9\u5206\u6790<\/strong>\uff1a\u30a2\u30af\u30c6\u30a3\u30d3\u30c6\u30a3\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u5185\u306e\u6700\u9577\u30d1\u30b9\u3092\u7279\u5b9a\u3057\u3001\u30d7\u30ed\u30b8\u30a7\u30af\u30c8\u5b8c\u4e86\u307e\u3067\u306e\u6700\u77ed\u6642\u9593\u3092\u6c7a\u5b9a\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u306a\u9055\u3044<\/strong><br><strong>\u30d1\u30b9 vs. \u30c8\u30ec\u30a4\u30eb<\/strong>\uff1a\u30d1\u30b9\u306f\u9802\u70b9\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002\u30c8\u30ec\u30a4\u30eb\u306f\u8fba\u3092\u7e70\u308a\u8fd4\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u305b\u3093\u3002<br><strong>\u30b5\u30fc\u30ad\u30c3\u30c8 vs. \u30b5\u30a4\u30af\u30eb<\/strong>\uff1a\u30b5\u30fc\u30ad\u30c3\u30c8\u306f\u9802\u70b9\u306e\u7e70\u308a\u8fd4\u3057\u304c\u53ef\u80fd\u3067\u3059\u304c\u3001\u30b5\u30a4\u30af\u30eb\u306f\uff08\u59cb\u70b9\u3068\u7d42\u70b9\u3092\u9664\u3044\u3066\uff09\u9802\u70b9\u306e\u7e70\u308a\u8fd4\u3057\u306f\u3067\u304d\u307e\u305b\u3093\u3002<br><strong>\u30aa\u30a4\u30e9\u30fc vs. \u30cf\u30df\u30eb\u30c8\u30f3<\/strong>\uff1a\u30aa\u30a4\u30e9\u30fc\u306f\u3059\u3079\u3066\u306e<strong>\u8fba<\/strong>\u3092\u4f7f\u7528\u3059\u308b\u3053\u3068\u3092\u6271\u3044\u307e\u3059\u3002\u30cf\u30df\u30eb\u30c8\u30f3\u306f\u3059\u3079\u3066\u306e<strong>\u9802\u70b9<\/strong>\u3092\u8a2a\u554f\u3059\u308b\u3053\u3068\u3092\u6271\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u96a3\u63a5\u884c\u5217<\/strong><br>QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c84\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u6c7a\u5b9a\u6570\u5b66\uff09\u306b\u304a\u3044\u3066\u3001\u96a3\u63a5\u884c\u5217\u306f\u30b0\u30e9\u30d5\u307e\u305f\u306f\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3092\u6570\u5024\u7684\u306b\u8868\u3059\u305f\u3081\u306b\u4f7f\u7528\u3055\u308c\u308b\u6b63\u65b9\u884c\u5217(<math data-latex=\"n\\times n\"><semantics><mrow><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n\\times n<\/annotation><\/semantics><\/math>)\u3067\u3059\u3002\u3053\u308c\u306f\u9802\u70b9\u9593\u306e\u76f4\u63a5\u63a5\u7d9a\uff08\u8fba\uff09\u306e\u6570\u3092\u8a18\u9332\u3057\u3001\u884c\u306f\u300c\u958b\u59cb\u300d\u9802\u70b9\u3001\u5217\u306f\u300c\u7d42\u4e86\u300d\u9802\u70b9\u3092\u8868\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u306a\u7279\u5fb4\u3068\u5b9a\u7fa9<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u69cb\u9020<\/strong>\uff1a(<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>)\u500b\u306e\u9802\u70b9\u3092\u6301\u3064\u30b0\u30e9\u30d5\u306e\u5834\u5408\u3001\u96a3\u63a5\u884c\u5217\u306f(<math data-latex=\"n\\times n\"><semantics><mrow><mi>n<\/mi><mo>\u00d7<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n\\times n<\/annotation><\/semantics><\/math>)\u3067\u3059\u3002\u3053\u306e\u884c\u5217\u306f\u3001\u884c\u3068\u5217\u304c\u540c\u3058\u9802\u70b9\u306e\u96c6\u5408\u306b\u540c\u3058\u9806\u5e8f\u3067\u5bfe\u5fdc\u3059\u308b\u3088\u3046\u306b\u914d\u7f6e\u3055\u308c\u307e\u3059\u3002<br><strong>\u5024<\/strong>\uff1a\u8981\u7d20(A_{ij})\u306f\u3001\u9802\u70b9(i)\u304b\u3089\u9802\u70b9(j)\u3078\u306e\u76f4\u63a5\u8fba\u306e\u6570\u3092\u793a\u3057\u307e\u3059\u3002<br><strong>\u5358\u7d14\u30b0\u30e9\u30d5<\/strong>: \u5358\u7d14\u30b0\u30e9\u30d5\uff08\u30eb\u30fc\u30d7\u3084\u591a\u91cd\u8fba\u304c\u306a\u3044\u30b0\u30e9\u30d5\uff09\u306e\u5834\u5408\u3001\u884c\u5217\u306f0\u30681\u306e\u307f\u3067\u69cb\u6210\u3055\u308c\u3001\u4e3b\u5bfe\u89d2\u7dda\u4e0a\u306b\u306f0\u304c\u5b58\u5728\u3057\u307e\u3059\u3002<br><strong>\u7121\u5411\u30b0\u30e9\u30d5<\/strong>: \u8fba\u304c\u53cc\u65b9\u5411\u3067\u3042\u308b\u305f\u3081\u3001\u884c\u5217\u306f\u4e3b\u5bfe\u89d2\u7dda\u306b\u95a2\u3057\u3066\u5bfe\u79f0\u3067\u3059\uff08\u3064\u307e\u308a\u3001(A_{ij}=A_{ji})\uff09\u3002<br><strong>\u6709\u5411\u30b0\u30e9\u30d5<\/strong>\uff08\u6709\u5411\u30b0\u30e9\u30d5\uff09\uff1a\u8fba\u306b\u306f\u7279\u5b9a\u306e\u65b9\u5411\u304c\u3042\u308b\u305f\u3081\uff08(i)\u304b\u3089(j)\u3078\u306e\u65b9\u5411\u306f(j)\u304b\u3089(i)\u3078\u306e\u65b9\u5411\u3068\u306f\u9650\u3089\u306a\u3044\u305f\u3081\uff09\u3001\u884c\u5217\u306f\u4e00\u822c\u306b\u5bfe\u79f0\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002<br><strong>\u30eb\u30fc\u30d7<\/strong>: \u9802\u70b9\u306b\u30eb\u30fc\u30d7\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u4e3b\u5bfe\u89d2\u7dda\u4e0a\u306b1\uff08\u6163\u4f8b\u306b\u3088\u3063\u3066\u306f2\u306e\u5834\u5408\u3082\u3042\u308a\u307e\u3059\uff09\u304c\u793a\u3055\u308c\u307e\u3059\u3002\u305f\u3060\u3057\u3001\u6a19\u6e96\u7684\u306a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u8868\u73fe\u3067\u306f\u3001\u30eb\u30fc\u30d7\u306f1\u3064\u306e\u8fba\u3068\u3057\u3066\u30ab\u30a6\u30f3\u30c8\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84\u306e\u5fdc\u7528<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u6b21\u6570\u306e\u6c7a\u5b9a<\/strong>\uff1a\u96a3\u63a5\u884c\u5217\u306e\u884c\uff08\u307e\u305f\u306f\u5217\uff09\u306e\u8981\u7d20\u306e\u548c\u306f\u3001\u5bfe\u5fdc\u3059\u308b\u9802\u70b9\u306e\u6b21\u6570\u3092\u8868\u3057\u307e\u3059\u3002\u6ce8\uff1a\u30eb\u30fc\u30d7\u304c\u5b58\u5728\u3059\u308b\u5834\u5408\u3001\u6b21\u6570\u306e\u8a08\u7b97\u6642\u306b\u30eb\u30fc\u30d7\u304c2\u56de\u30ab\u30a6\u30f3\u30c8\u3055\u308c\u308b\u3053\u3068\u304c\u3088\u304f\u3042\u308a\u307e\u3059\u3002<br>\u30d1\u30b9\u306e\u6570\u306e\u30ab\u30a6\u30f3\u30c8 (<math data-latex=\"M^{n}\"><semantics><msup><mi>M<\/mi><mi>n<\/mi><\/msup><annotation encoding=\"application\/x-tex\">M^{n}<\/annotation><\/semantics><\/math>)\uff1a\u884c\u5217\u306e\u3079\u304d\u4e57\uff08<math data-latex=\"A^{2}\"><semantics><msup><mi>A<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">A^{2}<\/annotation><\/semantics><\/math>,<math data-latex=\"A^{3}\"><semantics><msup><mi>A<\/mi><mn>3<\/mn><\/msup><annotation encoding=\"application\/x-tex\">A^{3}<\/annotation><\/semantics><\/math>\u306a\u3069\uff09\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u3001\u9802\u70b9\u9593\u306e\u7279\u5b9a\u306e\u9577\u3055\u306e\u30d1\u30b9\u306e\u6570\u3092\u6c7a\u5b9a\u3067\u304d\u307e\u3059\u3002<br><math data-latex=\"A^{2}\"><semantics><msup><mi>A<\/mi><mn>2<\/mn><\/msup><annotation encoding=\"application\/x-tex\">A^{2}<\/annotation><\/semantics><\/math>\uff1a\u306e\u8981\u7d20(<math data-latex=\"i,j\"><semantics><mrow><mi>i<\/mi><mo separator=\"true\">,<\/mo><mi>j<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">i,j<\/annotation><\/semantics><\/math>)\u306f\u3001\u9802\u70b9(<math data-latex=\"i\"><semantics><mi>i<\/mi><annotation encoding=\"application\/x-tex\">i<\/annotation><\/semantics><\/math>)\u304b\u3089\u9802\u70b9(<math data-latex=\"j\"><semantics><mi>j<\/mi><annotation encoding=\"application\/x-tex\">j<\/annotation><\/semantics><\/math>)\u3078\u306e2\u6bb5\u968e\u30d1\u30b9\uff082\u3064\u306e\u8fba\u3092\u6301\u3064\u30d1\u30b9\uff09\u306e\u6570\u3092\u793a\u3057\u307e\u3059\u3002<br>\u9023\u7d50\u6027\u306e\u8b58\u5225\uff1a\u96a3\u63a5\u884c\u5217\u306f\u3001\u30b0\u30e9\u30d5\u304c\u9023\u7d50\u3055\u308c\u3066\u3044\u308b\u304b\u3069\u3046\u304b\u3092\u5224\u65ad\u3059\u308b\u305f\u3081\u306b\u4f7f\u7528\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4f8b: \u7121\u5411\u30b0\u30e9\u30d5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9802\u70b9 {A, B, C} \u3092\u6301\u3064\u7121\u5411\u30b0\u30e9\u30d5\u3067\u3001A \u304c B \u306b\u63a5\u7d9a\u3057\u3001B \u304c C \u306b\u63a5\u7d9a\u3059\u308b\u5834\u5408:      \u3053\u3053\u3067\u3001\u884c A\/\u5217 B \u306f 1\u3001\u884c B\/\u5217 A \u306f 1 (\u5bfe\u79f0) \u3067\u3059\u3002<math data-latex=\"\\begin{matrix}&amp;A&amp;B&amp;C\\\\ A&amp;0&amp;1&amp;0\\\\ B&amp;1&amp;0&amp;1\\\\ C&amp;0&amp;1&amp;0\\end{matrix}\"><semantics><mtable columnalign=\"center center center center\"><mtr><mtd style=\"padding-left:0em;\"><mrow><\/mrow><\/mtd><mtd><mi>A<\/mi><\/mtd><mtd><mi>B<\/mi><\/mtd><mtd style=\"padding-right:0em;\"><mi>C<\/mi><\/mtd><\/mtr><mtr><mtd style=\"padding-left:0em;\"><mi>A<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd style=\"padding-right:0em;\"><mn>0<\/mn><\/mtd><\/mtr><mtr><mtd style=\"padding-left:0em;\"><mi>B<\/mi><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd style=\"padding-right:0em;\"><mn>1<\/mn><\/mtd><\/mtr><mtr><mtd style=\"padding-left:0em;\"><mi>C<\/mi><\/mtd><mtd><mn>0<\/mn><\/mtd><mtd><mn>1<\/mn><\/mtd><mtd style=\"padding-right:0em;\"><mn>0<\/mn><\/mtd><\/mtr><\/mtable><annotation encoding=\"application\/x-tex\">\\begin{matrix}&amp;A&amp;B&amp;C\\\\ A&amp;0&amp;1&amp;0\\\\ B&amp;1&amp;0&amp;1\\\\ C&amp;0&amp;1&amp;0\\end{matrix}<\/annotation><\/semantics><\/math><br>\u4f8b: \u6709\u5411\u30b0\u30e9\u30d5<br>\u8fba\u304c A \u304b\u3089 B \u3078\u306e\u307f\u4f38\u3073\u308b\u5834\u5408 (A <math data-latex=\"\\rightarrow\"><semantics><mo stretchy=\"false\" lspace=\"0em\" rspace=\"0em\">\u2192<\/mo><annotation encoding=\"application\/x-tex\">\\rightarrow<\/annotation><\/semantics><\/math> B)\u3001(<math data-latex=\"A_{AB}=1\"><semantics><mrow><msub><mi>A<\/mi><mrow><mi>A<\/mi><mi>B<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">A_{AB}=1<\/annotation><\/semantics><\/math>) \u3067\u3059\u304c\u3001(<math data-latex=\"A_{BA}=0\"><semantics><mrow><msub><mi>A<\/mi><mrow><mi>B<\/mi><mi>A<\/mi><\/mrow><\/msub><mo>=<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">A_{BA}=0<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002<\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u6709\u5411\u30b0\u30e9\u30d5<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">QLD 12\u5e74\u751f\u4e00\u822c\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c84\uff1a\u30cd\u30c3\u30c8\u30ef\u30fc\u30af\u3068\u6c7a\u5b9a\u6570\u5b66\uff09\u3067\u306f\u3001\u6709\u5411\u30b0\u30e9\u30d5\uff08Directed Graph\u306e\u7565\uff09\u3068\u306f\u3001\u8fba\u304c\u7279\u5b9a\u306e\u65b9\u5411\u3092\u6301\u3064\u30b0\u30e9\u30d5\u306e\u4e00\u7a2e\u3067\u3059\u3002\u6a19\u6e96\u7684\u306a\u30b0\u30e9\u30d5\u306e\u3088\u3046\u306b\u5358\u7d14\u306a\u53cc\u65b9\u5411\u306e\u63a5\u7d9a\u3092\u8868\u3059\u306e\u3067\u306f\u306a\u304f\u3001\u6709\u5411\u30b0\u30e9\u30d5\u306e\u8fba\u306f\u77e2\u5370\u3067\u8868\u3055\u308c\u3001\u4e00\u65b9\u5411\u306e\u95a2\u4fc2\u3092\u793a\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u4e3b\u8981\u306a\u69cb\u6210\u8981\u7d20\u3068\u7279\u5fb4<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u9802\u70b9\uff08\u30ce\u30fc\u30c9\uff09\uff1a\u70b9\u307e\u305f\u306f\u70b9\u3067\u3001\u7269\u4f53\u307e\u305f\u306f\u4f4d\u7f6e\u3092\u8868\u3057\u307e\u3059\u3002<br>\u5f27\uff08\u6709\u5411\u8fba\uff09\uff1a\u9802\u70b9\u3092\u7d50\u3076\u77e2\u5370\u3002\u65b9\u5411\u306f\u91cd\u8981\u3067\u3042\u308a\u3001\u4f8b\u3048\u3070\u300c\u30b5\u30e1\u304c\u9b5a\u3092\u98df\u3079\u308b\u300d\uff08\u4e00\u65b9\u5411\uff09\u3067\u306f\u306a\u304f\u3001\u300c\u30b5\u30e1\u3068\u9b5a\u306f\u5883\u754c\u7dda\u3092\u5171\u6709\u3057\u3066\u3044\u308b\u300d\u3092\u8868\u3057\u307e\u3059\u3002<br>\u9806\u5e8f\u4ed8\u304d\u30da\u30a2\uff1a\u5404\u5f27 ((u,v)) \u306f\u3001\u30bd\u30fc\u30b9 ((u)) \u304b\u3089\u30bf\u30fc\u30b2\u30c3\u30c8 ((v)) \u306b\u5411\u304b\u3046\u3001\u9806\u5e8f\u4ed8\u304d\u9802\u70b9\u306e\u30da\u30a2\u3067\u3059\u3002<br>\u5358\u7d14\u6709\u5411\u30b0\u30e9\u30d5: \u30eb\u30fc\u30d7\uff08\u540c\u3058\u9802\u70b9\u304b\u3089\u59cb\u307e\u308a\u3001\u540c\u3058\u9802\u70b9\u304b\u3089\u7d42\u308f\u308b\u8fba\uff09\u304c\u306a\u304f\u3001\u540c\u30582\u3064\u306e\u9802\u70b9\u9593\u306b\u8907\u6570\u306e\u8fba\u304c\u306a\u3044\u6709\u5411\u30b0\u30e9\u30d5\u3002<br>\u91cd\u307f\u4ed8\u304d\u6709\u5411\u30b0\u30e9\u30d5: \u5404\u77e2\u5370\u306b\u6570\u5024\uff08\u91cd\u307f\uff09\u304c\u5272\u308a\u5f53\u3066\u3089\u308c\u3001\u8ddd\u96e2\u3001\u6642\u9593\u3001\u30b3\u30b9\u30c8\u306a\u3069\u306e\u6307\u6a19\u3092\u8868\u3059\u6709\u5411\u30b0\u30e9\u30d5\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30e6\u30cb\u30c3\u30c84\u306e\u4e3b\u8981\u6982\u5ff5<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5165\u6b21\u6570: \u9802\u70b9\u3092\u6307\u3059\u77e2\u5370\u306e\u6570\u3002<br>\u51fa\u6b21\u6570: \u9802\u70b9\u304b\u3089\u5916\u306b\u5411\u304b\u3046\u77e2\u5370\u306e\u6570\u3002<br>\u9023\u7d50\u6027: \u3059\u3079\u3066\u306e\u77e2\u5370\u3092\u76f4\u7dda\u306b\u7f6e\u304d\u63db\u3048\u305f\u3068\u304d\u306b\u3001\u57fa\u3068\u306a\u308b\u30b0\u30e9\u30d5\u304c\u9023\u7d50\u3055\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u6709\u5411\u30b0\u30e9\u30d5\u306f\u300c\u9023\u7d50\u300d\uff08\u307e\u305f\u306f\u5f31\u9023\u7d50\uff09\u3055\u308c\u307e\u3059\u3002\u3059\u3079\u3066\u306e\u9802\u70b9\u304b\u3089\u4ed6\u306e\u3059\u3079\u3066\u306e\u9802\u70b9\u3078\u306e\u6709\u5411\u30d1\u30b9\u304c\u3042\u308b\u5834\u5408\u3001\u6709\u5411\u30b0\u30e9\u30d5\u306f\u300c\u5f37\u9023\u7d50\u300d\u3055\u308c\u307e\u3059\u3002<br>\u8868\u73fe: 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class=\"wp-block-paragraph\">\u307e\u3068\u3081\u308b\u3068\u3001QLD\u4e00\u822c\u6570\u5b66\u306b\u304a\u3051\u308b\u6709\u5411\u30b0\u30e9\u30d5\u306f\u3001\u95a2\u4fc2\u3001\u30d5\u30ed\u30fc\u3001\u307e\u305f\u306f\u30d1\u30b9\u306e\u65b9\u5411\u304c\u91cd\u8981\u306a\u30b7\u30b9\u30c6\u30e0\u306e\u30e2\u30c7\u30ea\u30f3\u30b0\u306b\u91cd\u70b9\u3092\u7f6e\u3044\u3066\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">********************************************************************************************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In practical financial applications and graph theory. It covers compound interest, loans, annuities, and perpetuity calculations (Topic 1) , alongside network concepts such as graphs, paths, trails, and,Eulerian\/Hamiltonian graphs for modeling decision-making (Topic 2).&nbsp; Key Topics in Unit 4: General Mathematics&nbsp; This unit prepares students for real-world financial management and using networks to solve optimization [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-1297","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1297","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1297"}],"version-history":[{"count":10,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1297\/revisions"}],"predecessor-version":[{"id":1363,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1297\/revisions\/1363"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1297"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1297"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1297"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}