Question

Let $R$ be the set of real numbers and let $G\subseteq R^2$ be a relation defined by
$G=\{(a,b),(c,d)|b-a=d-c\}.$ . Then, $G$ is

• A. reflexive only
• B. symmetric only
• C. transitive only
• D. an equivalence relation

Answer 1

Answer: (D)

Given, R is the set of real numbers. A relation defined by
$G=\{(a,b),(c,d):b-a=d-c\}$ and $G\subseteq R^2$ For reflexive $(a,b)G(a,b)$
$\Rightarrow$ $b-a=b-a$
$\Rightarrow$ G is reflexive For symmetric $(a,b)G(c,d)$
$\Rightarrow$ $b-a=d-c$
$\Rightarrow$ $d-c=b-a$
$\Rightarrow$ $(c,d)G(a,b)$
$\Rightarrow$ G is symmetric. For transitive $\{(a,b),(c,d)\}\in G$ and $\{(c,d),(e,f)\}\in G$
$\Rightarrow$ $b-a=d-c$ and $d-c=f-e$
$\Rightarrow$ $b-a=f-e$
$\Rightarrow$ $\{(a,b),(e,f)\}\in G$
$\Rightarrow$ G is transitive. So, G is reflexive, symmetric and transitive. Hence, G is an equivalence relation.