{"id":1204,"date":"2026-01-21T20:20:45","date_gmt":"2026-01-21T10:20:45","guid":{"rendered":"https:\/\/archive4ones.com\/2ndstudy\/?p=1204"},"modified":"2026-01-21T21:26:42","modified_gmt":"2026-01-21T11:26:42","slug":"year11-math-4-1-3-introduction-to-proof","status":"publish","type":"post","link":"https:\/\/archive4ones.com\/2ndstudy\/?p=1204","title":{"rendered":"Year11- MATH-4-1-3 Introduction to Proof"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">&#8220;Introduction to Proof &#8211; Circle geometry, geometric proof, and the beginning of formal mathematical logic&#8221; is a foundational topic in&nbsp;<strong>Unit 1 of the Queensland Curriculum &amp; Assessment Authority (QCAA) Specialist Mathematics Syllabus<\/strong>&nbsp;for Grade 11. This subject&nbsp;<mark>serves as the bridge between computational mathematics and formal, abstract reasoning, focusing on proving why geometric properties are true rather than just applying formulas<\/mark>.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">It is typically covered within &#8220;Topic 2: Introduction to proof&#8221; and &#8220;Topic 3 &amp; 4: Geometry in the plane and proof&#8221; (often combined in resources).&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Components of this Unit&nbsp;<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">The topic introduces students to the rigorous structure of mathematical arguments and applies these techniques specifically to circular shapes.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Beginning of Formal Mathematical Logic:<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Terminology and Notation:<\/strong> Students learn the language of proofs, including symbols for &#8220;for all&#8221; (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo>\u2200<\/mo><annotation encoding=\"text\/plain\">for all<\/annotation><\/semantics><\/math>), &#8220;there exists&#8221; (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo>\u2203<\/mo><annotation encoding=\"text\/plain\">there exists<\/annotation><\/semantics><\/math>), &#8220;implies&#8221; (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo>\u21d2<\/mo><annotation encoding=\"text\/plain\">implies<\/annotation><\/semantics><\/math>), and &#8220;if and only if&#8221; (<math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mo>\u27fa<\/mo><annotation encoding=\"text\/plain\">\u27fa<\/annotation><\/semantics><\/math>).<\/li>\n\n\n\n<li><strong>Methods of Proof:<\/strong> Direct proof, proof by contradiction, and identifying the converse of a statement.<\/li>\n\n\n\n<li><strong>Number Systems:<\/strong> Understanding the properties of rational and irrational numbers.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Circle Geometry (Circle Theorems):<\/strong>\n<ul class=\"wp-block-list\">\n<li><strong>Angle Properties:<\/strong> Angles in a semicircle, angles at the centre, angles in the same segment, and opposite angles of a cyclic quadrilateral.<\/li>\n\n\n\n<li><strong>Chords and Tangents:<\/strong> Properties of lines drawn from the center of a circle perpendicular to a chord, and properties of tangent lines to a circle.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Geometric Proof:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Using congruent triangles (SSS, SAS, AAS, RHS) and similarity tests to prove circle theorems.<\/li>\n\n\n\n<li>Constructing logical arguments where each step is justified by a known reason (e.g., &#8220;radii equal&#8221;, &#8220;angle at center&#8221;).&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Why is this Taught?&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This unit is designed to shift students from &#8220;how to get the answer&#8221; to &#8220;why the answer is correct&#8221;. It develops logical, deductive reasoning skills required for advanced engineering or mathematics studies at university level.&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Assessment Focus:<\/strong> Formal proofs are a key part of the QCE (Queensland Certificate of Education) Specialist Mathematics syllabus, with specific questions testing the ability to construct, justify, and write proofs.<\/li>\n\n\n\n<li><strong>Context in Unit 1:<\/strong> It follows combinatorics and accompanies vector geometry in Unit 1, setting the stage for more complex proofs in Units 3 and 4 (such as mathematical induction).&nbsp;<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">****************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Year 11 Specialist Mathematics in Queensland (Unit 1, Topic 2) introduces formal proof techniques, including direct proof, contradiction, and circle theorems. Common problems involve&nbsp;<mark>algebraic proofs (e.g., consecutive integers), geometric properties (e.g., tangents and radii), and logical deductions to verify mathematical statements<\/mark>.&nbsp;<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Key Sample Problems and Solutions<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Algebraic Proof (Direct):<\/strong>\n<ul class=\"wp-block-list\">\n<li><em>Problem:<\/em> Prove that if <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mi>y<\/mi><\/mrow><annotation encoding=\"text\/plain\">x is greater than y<\/annotation><\/semantics><\/math>, then <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&gt;<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"text\/plain\">x plus c is greater than y plus c<\/annotation><\/semantics><\/math>.<\/li>\n\n\n\n<li><em>Solution:<\/em> Assume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mi>y<\/mi><\/mrow><annotation encoding=\"text\/plain\">x is greater than y<\/annotation><\/semantics><\/math>. This implies <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">x minus y is greater than 0<\/annotation><\/semantics><\/math>. Let <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mi>k<\/mi><\/mrow><annotation encoding=\"text\/plain\">x minus y equals k<\/annotation><\/semantics><\/math>, where <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>k<\/mi><annotation encoding=\"text\/plain\">k<\/annotation><\/semantics><\/math> is a positive number. Then (\ud835\udc65+\ud835\udc50)\u2212(\ud835\udc66+\ud835\udc50)=\ud835\udc65+\ud835\udc50\u2212\ud835\udc66\u2212\ud835\udc50=\ud835\udc65\u2212\ud835\udc66=\ud835\udc58. Since <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">k is greater than 0<\/annotation><\/semantics><\/math>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>)<\/mo><mo>\u2212<\/mo><mo>(<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><mo>)<\/mo><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"text\/plain\">open paren x plus c close paren minus open paren y plus c close paren is greater than 0<\/annotation><\/semantics><\/math>, therefore <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&gt;<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"text\/plain\">x plus c is greater than y plus c<\/annotation><\/semantics><\/math>.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Algebraic Proof (Number Theory):<\/strong>\n<ul class=\"wp-block-list\">\n<li><em>Problem:<\/em> Prove that the sum of the squares of any two consecutive integers is always one more than a multiple of 4.<\/li>\n\n\n\n<li><em>Solution:<\/em> Let the integers be <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math> and <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">n plus 1<\/annotation><\/semantics><\/math>. The sum of their squares is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo>)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">n squared plus open paren n plus 1 close paren squared equals n squared plus n squared plus 2 n plus 1 equals 2 n squared plus 2 n plus 1<\/annotation><\/semantics><\/math>. This can be written as <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo>(<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><mo>)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">2 open paren n squared plus n close paren plus 1<\/annotation><\/semantics><\/math>, which is not necessarily a multiple of 4. <em>Corrected Proposition:<\/em> Prove the sum is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">4 k plus 1<\/annotation><\/semantics><\/math> or <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"text\/plain\">4 k plus 2<\/annotation><\/semantics><\/math>. Actually, the sum of squares of two consecutive integers is <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo>(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo>)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">n squared plus open paren n plus 1 close paren squared equals 2 n squared plus 2 n plus 1<\/annotation><\/semantics><\/math>. For any integer <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mi>n<\/mi><annotation encoding=\"text\/plain\">n<\/annotation><\/semantics><\/math>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"text\/plain\">n squared plus n<\/annotation><\/semantics><\/math> is always even, so <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>2<\/mn><mo>(<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><mo>)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">2 open paren n squared plus n close paren plus 1<\/annotation><\/semantics><\/math> is always odd and specifically <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"text\/plain\">4 k plus 1<\/annotation><\/semantics><\/math> (e.g., <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>1<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>2<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"text\/plain\">1 squared plus 2 squared equals 5<\/annotation><\/semantics><\/math>, <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><msup><mn>2<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"text\/plain\">2 squared plus 3 squared equals 13<\/annotation><\/semantics><\/math>).<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Geometric Proof (Circle Properties):<\/strong>\n<ul class=\"wp-block-list\">\n<li><em>Problem:<\/em> Prove that a tangent drawn to a circle is perpendicular to the radius at the point of contact.<\/li>\n\n\n\n<li><em>Solution:<\/em> Uses proof by contradiction. Assume the tangent is not perpendicular. If so, a shortest distance line (perpendicular) can be drawn to the tangent, which would mean another point on the tangent is inside the circle, contradicting the definition of a tangent.<\/li>\n<\/ul>\n<\/li>\n\n\n\n<li><strong>Properties to Prove:<\/strong>\n<ul class=\"wp-block-list\">\n<li>Angles at the circumference subtended by the same arc are equal.<\/li>\n\n\n\n<li>Opposite angles of a cyclic quadrilateral are supplementary.&nbsp;<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>Common Proof Structures<\/strong>&nbsp;<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><strong>Direct Proof:<\/strong> <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo>\u27f9<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"text\/plain\">cap P \u27f9 cap Q<\/annotation><\/semantics><\/math> (start with premises, follow logic to conclusion).<\/li>\n\n\n\n<li><strong>Contradiction:<\/strong> Assume <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u00ac<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"text\/plain\">logical not cap Q<\/annotation><\/semantics><\/math> and show it leads to a false statement.<\/li>\n\n\n\n<li><strong>Contrapositive:<\/strong> To prove <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mi>P<\/mi><mo>\u27f9<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"text\/plain\">cap P \u27f9 cap Q<\/annotation><\/semantics><\/math>, prove <math xmlns=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><semantics><mrow><mo>\u00ac<\/mo><mi>Q<\/mi><mo>\u27f9<\/mo><mo>\u00ac<\/mo><mi>P<\/mi><\/mrow><annotation encoding=\"text\/plain\">logical not cap Q \u27f9 logical not cap P<\/annotation><\/semantics><\/math>.\u00a0<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">***************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u300c\u8a3c\u660e\u5165\u9580 \u2015 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class=\"wp-block-paragraph\">\u3053\u306e\u79d1\u76ee\u306f\u901a\u5e38\u3001\u300c\u30c8\u30d4\u30c3\u30af2\uff1a\u8a3c\u660e\u5165\u9580\u300d\u304a\u3088\u3073\u300c\u30c8\u30d4\u30c3\u30af3\uff064\uff1a\u5e73\u9762\u5e7e\u4f55\u5b66\u3068\u8a3c\u660e\u300d\uff08\u591a\u304f\u306e\u5834\u5408\u3001\u30ea\u30bd\u30fc\u30b9\u3068\u3057\u3066\u7d71\u5408\u3055\u308c\u3066\u3044\u307e\u3059\uff09\u3067\u6271\u308f\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30e6\u30cb\u30c3\u30c8\u306e\u4e3b\u8981\u306a\u69cb\u6210\u8981\u7d20<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u30c8\u30d4\u30c3\u30af\u3067\u306f\u3001\u6570\u5b66\u7684\u8b70\u8ad6\u306e\u53b3\u5bc6\u306a\u69cb\u9020\u3092\u751f\u5f92\u306b\u7d39\u4ecb\u3057\u3001\u3053\u308c\u3089\u306e\u624b\u6cd5\u3092\u7279\u306b\u5186\u5f62\u306b\u9069\u7528\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f62\u5f0f\u6570\u7406\u8ad6\u7406\u5b66\u5165\u9580\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u7528\u8a9e\u3068\u8868\u8a18\u6cd5\uff1a\u751f\u5f92\u306f\u300c\u3059\u3079\u3066\u306e\u5834\u5408\u306b\u304a\u3044\u3066\u300d\uff08<math data-latex=\"\u2200\"><semantics><mi>\u2200<\/mi><annotation encoding=\"application\/x-tex\">\u2200<\/annotation><\/semantics><\/math>\uff09\u3001\u300c\u5b58\u5728\u3059\u308b\u300d\uff08\u2203\uff09\u3001\u300c\u301c\u3092\u542b\u610f\u3059\u308b\u300d\uff08\u21d2\uff09\u3001\u300c\u301c\u306e\u5834\u5408\u306b\u9650\u308a\u300d\uff08\u27fa\uff09\u306a\u3069\u306e\u8a18\u53f7\u3092\u542b\u3080\u8a3c\u660e\u8a00\u8a9e\u3092\u5b66\u3073\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8a3c\u660e\u65b9\u6cd5\uff1a\u76f4\u63a5\u8a3c\u660e\u3001\u80cc\u7406\u6cd5\u306b\u3088\u308b\u8a3c\u660e\u3001\u305d\u3057\u3066\u547d\u984c\u306e\u9006\u306e\u7279\u5b9a\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u6570\u4f53\u7cfb\uff1a\u6709\u7406\u6570\u3068\u7121\u7406\u6570\u306e\u6027\u8cea\u3092\u7406\u89e3\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5186\u306e\u5e7e\u4f55\u5b66\uff08\u5186\u306e\u5b9a\u7406\uff09\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u89d2\u306e\u6027\u8cea\uff1a\u534a\u5186\u306e\u89d2\u3001\u4e2d\u5fc3\u306e\u89d2\u3001\u540c\u4e00\u7dda\u5206\u306e\u89d2\u3001\u5186\u5468\u56db\u8fba\u5f62\u306e\u5bfe\u89d2\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5f26\u3068\u63a5\u7dda\uff1a\u5186\u306e\u4e2d\u5fc3\u304b\u3089\u5f26\u306b\u5782\u76f4\u306b\u5f15\u3044\u305f\u76f4\u7dda\u306e\u6027\u8cea\u3001\u305d\u3057\u3066\u5186\u306e\u63a5\u7dda\u306e\u6027\u8cea\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5e7e\u4f55\u5b66\u7684\u8a3c\u660e\uff1a<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5408\u540c\u306a\u4e09\u89d2\u5f62\uff08SSS\u3001SAS\u3001AAS\u3001RHS\uff09\u3068\u76f8\u4f3c\u6027\u691c\u5b9a\u3092\u7528\u3044\u3066\u5186\u5468\u5b9a\u7406\u3092\u8a3c\u660e\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5404\u30b9\u30c6\u30c3\u30d7\u304c\u65e2\u77e5\u306e\u7406\u7531\uff08\u4f8b\uff1a\u300c\u534a\u5f84\u304c\u7b49\u3057\u3044\u300d\u3001\u300c\u4e2d\u5fc3\u89d2\u300d\uff09\u306b\u3088\u3063\u3066\u6b63\u5f53\u5316\u3055\u308c\u308b\u8ad6\u7406\u7684\u8b70\u8ad6\u3092\u69cb\u7bc9\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u306a\u305c\u3053\u306e\u5358\u5143\u3092\u6559\u3048\u308b\u306e\u304b\uff1f<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u3053\u306e\u5358\u5143\u306f\u3001\u751f\u5f92\u306e\u7406\u89e3\u3092\u300c\u7b54\u3048\u3092\u5c0e\u304f\u65b9\u6cd5\u300d\u304b\u3089\u300c\u306a\u305c\u7b54\u3048\u304c\u6b63\u3057\u3044\u306e\u304b\u300d\u3078\u3068\u79fb\u884c\u3055\u305b\u308b\u3088\u3046\u306b\u8a2d\u8a08\u3055\u308c\u3066\u3044\u307e\u3059\u3002\u5927\u5b66\u30ec\u30d9\u30eb\u306e\u9ad8\u5ea6\u306a\u5de5\u5b66\u307e\u305f\u306f\u6570\u5b66\u306e\u7814\u7a76\u306b\u5fc5\u8981\u306a\u3001\u8ad6\u7406\u7684\u304b\u3064\u6f14\u7e79\u7684\u306a\u63a8\u8ad6\u30b9\u30ad\u30eb\u3092\u990a\u3044\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8a55\u4fa1\u306e\u7126\u70b9\uff1a\u5f62\u5f0f\u7684\u8a3c\u660e\u306f\u3001QCE\uff08\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u6559\u80b2\u8cc7\u683c\uff09\u5c02\u9580\u6570\u5b66\u30b7\u30e9\u30d0\u30b9\u306e\u91cd\u8981\u306a\u90e8\u5206\u3067\u3042\u308a\u3001\u5177\u4f53\u7684\u306a\u554f\u984c\u3092\u901a\u3057\u3066\u3001\u8a3c\u660e\u3092\u69cb\u7bc9\u3057\u3001\u6b63\u5f53\u5316\u3057\u3001\u8a18\u8ff0\u3059\u308b\u80fd\u529b\u304c\u8a66\u3055\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u5358\u51431\u306b\u304a\u3051\u308b\u6587\u8108\uff1a\u5358\u51431\u306e\u7d44\u5408\u305b\u8ad6\u3068\u30d9\u30af\u30c8\u30eb\u5e7e\u4f55\u5b66\u3092\u88dc\u8db3\u3057\u3001\u5358\u51433\u30684\u306e\u3088\u308a\u8907\u96d1\u306a\u8a3c\u660e\uff08\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u306a\u3069\uff09\u3078\u306e\u57fa\u790e\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">***************************************************************************<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u30af\u30a4\u30fc\u30f3\u30ba\u30e9\u30f3\u30c9\u5dde\u306e11\u5e74\u751f\u5c02\u9580\u6570\u5b66\uff08\u30e6\u30cb\u30c3\u30c81\u3001\u30c8\u30d4\u30c3\u30af2\uff09\u3067\u306f\u3001\u76f4\u63a5\u8a3c\u660e\u3001\u80cc\u7406\u6cd5\u3001\u5186\u5468\u5b9a\u7406\u306a\u3069\u306e\u6b63\u5f0f\u306a\u8a3c\u660e\u6280\u6cd5\u3092\u5b66\u3073\u307e\u3059\u3002\u3088\u304f\u51fa\u984c\u3055\u308c\u308b\u554f\u984c\u306b\u306f\u3001\u4ee3\u6570\u7684\u8a3c\u660e\uff08\u4f8b\uff1a\u9023\u7d9a\u6574\u6570\uff09\u3001\u5e7e\u4f55\u5b66\u7684\u6027\u8cea\uff08\u4f8b\uff1a\u63a5\u7dda\u3084\u534a\u5f84\uff09\u3001\u305d\u3057\u3066\u6570\u5b66\u7684\u547d\u984c\u3092\u691c\u8a3c\u3059\u308b\u305f\u3081\u306e\u8ad6\u7406\u7684\u6f14\u7e79\u304c\u542b\u307e\u308c\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e3b\u306a\u30b5\u30f3\u30d7\u30eb\u554f\u984c\u3068\u89e3\u7b54<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4ee3\u6570\u7684\u8a3c\u660e\uff08\u76f4\u63a5\uff09<\/strong>\uff1a\u554f\u984c\uff1a(<math data-latex=\"x&gt;y\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x&gt;y<\/annotation><\/semantics><\/math>) \u306a\u3089\u3070 (<math data-latex=\"x+c&gt;y+c\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&gt;<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x+c&gt;y+c<\/annotation><\/semantics><\/math>) \u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u306a\u3055\u3044\u3002\u89e3\u7b54\uff1a(<math data-latex=\"x&gt;y\"><semantics><mrow><mi>x<\/mi><mo>&gt;<\/mo><mi>y<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x&gt;y<\/annotation><\/semantics><\/math>) \u3068\u4eee\u5b9a\u3059\u308b\u3002\u3053\u308c\u306f (<math data-latex=\"x-y&gt;0\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">x-y&gt;0<\/annotation><\/semantics><\/math>) \u3092\u610f\u5473\u3059\u308b\u3002(<math data-latex=\"x-y=k\"><semantics><mrow><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x-y=k<\/annotation><\/semantics><\/math>) \u3068\u3059\u308b\u3002\u3053\u3053\u3067 (<math data-latex=\"k\"><semantics><mi>k<\/mi><annotation encoding=\"application\/x-tex\">k<\/annotation><\/semantics><\/math>) \u306f\u6b63\u306e\u6570\u3067\u3042\u308b\u3002\u3059\u308b\u3068 <math data-latex=\"(x+c)-(y+c)=x+c-y-c=x-y=k\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>=<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>\u2212<\/mo><mi>c<\/mi><mo>=<\/mo><mi>x<\/mi><mo>\u2212<\/mo><mi>y<\/mi><mo>=<\/mo><mi>k<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">(x+c)-(y+c)=x+c-y-c=x-y=k<\/annotation><\/semantics><\/math> \u3068\u306a\u308b\u3002 (<math data-latex=\"k&gt;0\"><semantics><mrow><mi>k<\/mi><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">k&gt;0<\/annotation><\/semantics><\/math>)\u3001(<math data-latex=\"(x+c)-(y+c)&gt;0\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2212<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>&gt;<\/mo><mn>0<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">(x+c)-(y+c)&gt;0<\/annotation><\/semantics><\/math>) \u306a\u306e\u3067\u3001(<math data-latex=\"x+c&gt;y+c\"><semantics><mrow><mi>x<\/mi><mo>+<\/mo><mi>c<\/mi><mo>&gt;<\/mo><mi>y<\/mi><mo>+<\/mo><mi>c<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">x+c&gt;y+c<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4ee3\u6570\u7684\u8a3c\u660e\uff08\u6570\u8ad6<\/strong>\uff09\uff1a\u554f\u984c\uff1a\u4efb\u610f\u306e2\u3064\u306e\u9023\u7d9a\u3059\u308b\u6574\u6570\u306e\u5e73\u65b9\u548c\u306f\u3001\u5e38\u306b4\u306e\u500d\u6570\u3088\u308a1\u5927\u304d\u3044\u3053\u3068\u3092\u8a3c\u660e\u3057\u306a\u3055\u3044\u3002\u89e3\u7b54\uff1a\u6574\u6570\u3092 (<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>) \u3068 (<math data-latex=\"n+1\"><semantics><mrow><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n+1<\/annotation><\/semantics><\/math>) \u3068\u3057\u307e\u3059\u3002\u3053\u308c\u3089\u306e\u5e73\u65b9\u548c\u306f (<math data-latex=\"n^{2}+(n+1)^{2}=n^{2}+n^{2}+2n+1=2n^{2}+2n+1\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><mo>=<\/mo><mn>2<\/mn><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n^{2}+(n+1)^{2}=n^{2}+n^{2}+2n+1=2n^{2}+2n+1<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002\u3053\u308c\u306f (<math data-latex=\"2(n^{2}+n)+1\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2(n^{2}+n)+1<\/annotation><\/semantics><\/math>) \u3068\u66f8\u304f\u3053\u3068\u3082\u3067\u304d\u307e\u3059\u304c\u3001\u5fc5\u305a\u3057\u30824\u306e\u500d\u6570\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u4fee\u6b63\u547d\u984c\uff1a\u548c\u304c (<math data-latex=\"4k+1\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4k+1<\/annotation><\/semantics><\/math>) \u307e\u305f\u306f (<math data-latex=\"4k+2\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>2<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4k+2<\/annotation><\/semantics><\/math>) \u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u306a\u3055\u3044\u3002\u5b9f\u969b\u3001\u9023\u7d9a\u3059\u308b2\u3064\u306e\u6574\u6570\u306e\u5e73\u65b9\u548c\u306f (<math data-latex=\"n^{2}+(n+1)^{2}=2n^{2}+2n+1\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mn>2<\/mn><\/msup><mo>=<\/mo><mn>2<\/mn><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mn>2<\/mn><mi>n<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">n^{2}+(n+1)^{2}=2n^{2}+2n+1<\/annotation><\/semantics><\/math>) \u3067\u3059\u3002\u4efb\u610f\u306e\u6574\u6570 (<math data-latex=\"n\"><semantics><mi>n<\/mi><annotation encoding=\"application\/x-tex\">n<\/annotation><\/semantics><\/math>) \u306b\u5bfe\u3057\u3066\u3001(<math data-latex=\"n^{2}+n\"><semantics><mrow><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">n^{2}+n<\/annotation><\/semantics><\/math>) \u306f\u5e38\u306b\u5076\u6570\u306a\u306e\u3067\u3001(<math data-latex=\"2(n^{2}+n)+1\"><semantics><mrow><mn>2<\/mn><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mi>n<\/mi><mn>2<\/mn><\/msup><mo>+<\/mo><mi>n<\/mi><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2(n^{2}+n)+1<\/annotation><\/semantics><\/math>) \u306f\u5e38\u306b\u5947\u6570\u3067\u3042\u308a\u3001\u5177\u4f53\u7684\u306b\u306f (<math data-latex=\"4k+1\"><semantics><mrow><mn>4<\/mn><mi>k<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">4k+1<\/annotation><\/semantics><\/math>) \u3067\u3059\uff08\u4f8b\uff1a(<math data-latex=\"1^{2}+2^{2}=5\"><semantics><mrow><msup><mn>1<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>2<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>5<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">1^{2}+2^{2}=5<\/annotation><\/semantics><\/math>)\u3001(<math data-latex=\"2^{2}+3^{2}=13\"><semantics><mrow><msup><mn>2<\/mn><mn>2<\/mn><\/msup><mo>+<\/mo><msup><mn>3<\/mn><mn>2<\/mn><\/msup><mo>=<\/mo><mn>13<\/mn><\/mrow><annotation encoding=\"application\/x-tex\">2^{2}+3^{2}=13<\/annotation><\/semantics><\/math>)\uff09\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u5e7e\u4f55\u5b66\u7684\u8a3c\u660e\uff08\u5186\u306e\u6027\u8cea<\/strong>\uff09\uff1a\u554f\u984c\uff1a\u5186\u306b\u5f15\u3044\u305f\u63a5\u7dda\u306f\u3001\u63a5\u70b9\u306b\u304a\u3044\u3066\u534a\u5f84\u306b\u5782\u76f4\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u306a\u3055\u3044\u3002\u89e3\u7b54\uff1a\u80cc\u7406\u6cd5\u3092\u7528\u3044\u307e\u3059\u3002\u63a5\u7dda\u306f\u5782\u76f4\u3067\u306f\u306a\u3044\u3068\u4eee\u5b9a\u3057\u307e\u3059\u3002\u3082\u3057\u305d\u3046\u306a\u3089\u3001\u63a5\u7dda\u306b\u6700\u77ed\u8ddd\u96e2\u306e\u7dda\uff08\u5782\u7dda\uff09\u3092\u5f15\u304f\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u304c\u3001\u3053\u308c\u306f\u63a5\u7dda\u4e0a\u306e\u5225\u306e\u70b9\u304c\u5186\u306e\u5185\u5074\u306b\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u3001\u63a5\u7dda\u306e\u5b9a\u7fa9\u306b\u77db\u76fe\u3057\u307e\u3059\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u8a3c\u660e\u3059\u3079\u304d\u6027\u8cea\uff1a\u540c\u3058\u5f27\u304c\u56f2\u3080\u5186\u5468\u4e0a\u306e\u89d2\u306f\u7b49\u3057\u3044\u3002\u5186\u5468\u56db\u8fba\u5f62\u306e\u53cd\u5bfe\u89d2\u306f\u88dc\u89d2\u3067\u3042\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><strong>\u4e00\u822c\u7684\u306a\u8a3c\u660e\u69cb\u9020<\/strong><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">\u76f4\u63a5\u8a3c\u660e\uff1a(<math data-latex=\"P\\implies Q\"><semantics><mrow><mi>P<\/mi><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P\\implies Q<\/annotation><\/semantics><\/math>)\uff08\u524d\u63d0\u304b\u3089\u59cb\u3081\u3066\u3001\u8ad6\u7406\u7684\u306b\u7d50\u8ad6\u307e\u3067\u9032\u3080\uff09\u3002<br>\u80cc\u7406\uff1a(<math data-latex=\"\\neg Q\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">\u00ac<\/mo><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\neg Q<\/annotation><\/semantics><\/math>) \u3092\u4eee\u5b9a\u3057\u3001\u305d\u308c\u304c\u8aa4\u3063\u305f\u547d\u984c\u306b\u3064\u306a\u304c\u308b\u3053\u3068\u3092\u793a\u3059\u3002<br>\u9006\u8a3c\u660e\uff1a(<math data-latex=\"P\\implies Q\"><semantics><mrow><mi>P<\/mi><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mi>Q<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">P\\implies Q<\/annotation><\/semantics><\/math>) \u3092\u8a3c\u660e\u3059\u308b\u306b\u306f\u3001(<math data-latex=\"\\neg Q\\implies \\neg P\"><semantics><mrow><mo form=\"prefix\" stretchy=\"false\">\u00ac<\/mo><mi>Q<\/mi><mspace width=\"0.2778em\"><\/mspace><mo stretchy=\"false\">\u27f9<\/mo><mspace width=\"0.2778em\"><\/mspace><mo form=\"prefix\" stretchy=\"false\">\u00ac<\/mo><mi>P<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">\\neg Q\\implies \\neg P<\/annotation><\/semantics><\/math>) \u3092\u8a3c\u660e\u3059\u308b\u3002<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8220;Introduction to Proof &#8211; Circle geometry, geometric proof, and the beginning of formal mathematical logic&#8221; is a foundational topic in&nbsp;Unit 1 of the Queensland Curriculum &amp; Assessment Authority (QCAA) Specialist Mathematics Syllabus&nbsp;for Grade 11. This subject&nbsp;serves as the bridge between computational mathematics and formal, abstract reasoning, focusing on proving why geometric properties are true rather [&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-1204","post","type-post","status-publish","format-standard","hentry","category-math"],"_links":{"self":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1204","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=1204"}],"version-history":[{"count":5,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1204\/revisions"}],"predecessor-version":[{"id":1256,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=\/wp\/v2\/posts\/1204\/revisions\/1256"}],"wp:attachment":[{"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1204"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1204"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/archive4ones.com\/2ndstudy\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1204"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}