<p>Let <em>R</em> = {(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)} be a relation on the set <em>A</em> = {1, 2, 3, 4}. The relation <em>R</em> is</p>
Step-by-Step Solution
Key Concept: A relation is transitive if whenever (a,b) and (b,c) are in R, then (a,c) must also be in R. Check this condition systematically for all pairs to determine if R is transitive, reflexive, or symmetric.
<p><strong>Step 1:</strong> Check if R is reflexive - need (1,1), (2,2), (3,3), (4,4) in R. Since (2,2) ∉ R, R is not reflexive.</p><p><strong>Step 2:</strong> Check if R is symmetric - for each (a,b) ∈ R, need (b,a) ∈ R. We have (1,3) ∈ R but (3,1) ∈ R ✓; (4,2) ∈ R but (2,4) ∈ R ✓; (2,3) ∈ R but (3,2) ∉ R ✗. So R is not symmetric.</p><p><strong>Step 3:</strong> Check if R is transitive - if (a,b), (b,c) ∈ R then (a,c) must be in R. Checking: (1,3)&(3,1)→need(1,1)✗; (4,2)&(2,4)→need(4,4)✗; (4,2)&(2,3)→need(4,3)✗; (2,4)&(4,2)→need(2,2)✗. R is not transitive.</p><p><strong>Step 4:</strong> R is neither reflexive, nor symmetric, nor transitive. It is simply a relation with no special properties.</p><p>∴ Answer: C (Neither reflexive, nor symmetric, nor transitive)</p>
Correct Answer: C