Sets, Relations & Functions
Types of Relations
Grade None
Question:
<p>Consider the following two binary relations on the set \(A = \{a, b, c\}\):<br>\(R_1 = \{(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)\}\) and<br>\(R_2 = \{(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)\}\).<br>Then:</p>
<p>both \(R_1\) and \(R_2\) are not symmetric.</p>
<p>\(R_1\) is not symmetric but it is transitive.</p>
<p>\(R_2\) is symmetric but it is not transitive.</p>
<p>both \(R_1\) and \(R_2\) are transitive.</p>
Step-by-Step Solution
Key Concept: A relation is symmetric if (x,y) ∈ R implies (y,x) ∈ R for all x,y. Check both directions systematically: R₁ must have (a,b) whenever (b,a) exists, and R₂ must satisfy the same property. Missing even one reciprocal pair breaks symmetry.
<p><strong>Step 1: Test R₁ for symmetry</strong></p><p>For R₁ to be symmetric, if (x,y) ∈ R₁ then (y,x) ∈ R₁.</p><p>Check (b,c) ∈ R₁: We need (c,b) ∈ R₁. But (c,b) ∉ R₁.</p><p>∴ R₁ is <strong>NOT symmetric</strong></p><p><strong>Step 2: Test R₂ for symmetry</strong></p><p>Check all pairs in R₂:</p><ul><li>(a,b) ∈ R₂ and (b,a) ∈ R₂ ✓</li><li>(a,c) ∈ R₂ and (c,a) ∈ R₂ ✓</li><li>(a,a), (b,b), (c,c) are reflexive (self-symmetric) ✓</li></ul><p>Every ordered pair has its symmetric counterpart in R₂.</p><p>∴ R₂ <strong>IS symmetric</strong></p><p><strong>Conclusion:</strong> The correct statement is that R₂ is symmetric but R₁ is not.</p><p>∴ Answer: C</p>
Correct Answer: C