Question

Let $R=\left\{ \left(3,3\right) \left(5,5\right), \left(9,9\right), \left(12,12\right), \left(5,12\right), \left(3,9\right), \left(3,12\right), \left(3,5\right)\right\}$ be a relation on the set $A=\left\{3,5,9,12\right\}$. Then, $R$ is:

• A. reflexive, symmetric but not transitive
• B. symmetric, transitive but not reflexive
• C. an equivalence relation
• D. reflexive, transitive but not symmetric

Answer 1

Answer: (D)

Let $R=\left\{ \left(3,3\right) \left(5,5\right), \left(9,9\right), \left(12,12\right), \left(5,12\right), \left(3,9\right), \left(3,12\right), \left(3,5\right)\right\}$ be a relation on the set $A=\left\{3,5,9,12\right\}$
Clearly, every element of A is related to itself. Therefore, it is a reflexive.
Now, $R$ is not symmetry because 3 is related to 5 but 5 is not related to 3.
Also $R$ is transitive relation because it satisfies the property that if a R b and b R c then $a\,R\,c.$