Sets, Relations & Functions
Equivalence Relations
Grade 11
Question:
<p>Let <em>T</em> be the set of all triangles in the Euclidean plane, and let a relation <em>R</em> on <em>T</em> be defined as <em>a R b</em> if <em>a</em> is congruent to <em>b</em> for all <em>a, b</em> ∈ <em>T</em>. Then <em>R</em> is</p>
<p>reflexive but not symmetric</p>
<p>transitive but not symmetric</p>
<p>equivalence relation</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: An equivalence relation must satisfy three properties simultaneously: reflexivity (every element relates to itself), symmetry (if a relates to b, then b relates to a), and transitivity (if a relates to b and b relates to c, then a relates to c). Congruence of triangles satisfies all three properties.
<p><strong>Step 1: Check Reflexivity</strong></p><p>Every triangle a is congruent to itself (a ≅ a). So a R a for all a ∈ T. ✓</p><p><strong>Step 2: Check Symmetry</strong></p><p>If a R b, then a ≅ b. By definition of congruence, b ≅ a, so b R a. ✓</p><p><strong>Step 3: Check Transitivity</strong></p><p>If a R b and b R c, then a ≅ b and b ≅ c. By the transitive property of congruence in Euclidean geometry, a ≅ c, so a R c. ✓</p><p><strong>Step 4: Conclusion</strong></p><p>Since R is reflexive, symmetric, and transitive, R is an equivalence relation on T. The congruence relation partitions T into equivalence classes of mutually congruent triangles.</p><p>∴ Answer: C (R is an equivalence relation)</p>
Correct Answer: C