Let A = 1,2,3 . Then, the number of relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive, is

Question

Let A = 1,2,3 . Then, the number of relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive, is (A) 1 (B) 2 (C) 3 (D) 4

Tardigrade Admin · Accepted Answer

Let R be a relation containing (1,2) and (1,3) R is reflexive, if (1,1),(2,2),(3,3) ∈ R Relation R is symmetric, if (2,1) ∈ R but (3,1) not ∈ R. But relation R is not transitive as (3,1),(1,2) ∈ R but (3,2) ∉ R Now, if we add the pair (3,2) and (2,3) to relation R, then relation R will become transitive. Hence, the total number of desired relations is one.

Q. Let A={1,2,3}. Then, the number of relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive, is

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Q. Let $A = {1, 2, 3} .$ Then, the number of relations containing $(1, 2)$ and $(1, 3)$ , which are reflexive and symmetric but not transitive, is