Question

Let R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.
(A) R is reflexive and symmetric but not transitive.
(B) R is reflexive and transitive but not symmetric.
(C) R is symmetric and transitive but not reflexive.
(D) R is an equivalence relation.

Answer 1

Here, R={(1,2),(2,2),(1,1),(4,4)(1,3),(3,3),(3,2)}
Since, $(a,a)\in R$., for every $a\in \{1,2,3,4\}$. Therefore, R is reflexive.
Now since $(1,2)\in R$ but $(2,1)\text{/}\in R$. Therefore, R is not symmetric.
Also, it is observed that $(a,b),(b,c)\in R$.
$\Rightarrow (a,c)\in R$. For all $a,b,c\in \{1,2,3,4\}$.
Therefore, R is transitive. Hence, R is reflexive and transitive but not symmetric. Thus, the correct answer is (b).