Question

Let $R_1$ be a relation defined by $R_1=\{(a, b) \mid a \geq b ; a, b \in R\}$. Then $R_1$ is

(A) An equivalence relation on $R$

(B) Reflexive, transitive but not symmetric

(C) Symmetric, Transitive but not reflexive

(D) Neither transitive nor reflexive but symmetric

Answer 1

For any $a \in R$, we have $a \geq a$, Therefore the relation $R$, is reflexive but it is not symmetric as $(2,1) \in R$, but $(1,2) \notin R_1$. 

The relation $R_1$ is transitive also, because $(a, b) \in R_1,(b, c) \in R_1$, imply that $a \geq b$ and $b \geq c$ which is turn imply that $a \geq c \Rightarrow(a, c) \in R_1$.

Above Correct Answer is Option B.