Check whether the relation R defined on the set A = 1, 2, 3, 4, 5… - Vidyadip

Question

Check whether the relation R defined on the set A = {1, 2, 3, 4, 5, 6} as R = {(a, b): b = a + 1} is reflexive, symmetric or transitive.

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Answer

Reflexivity: Let a be an arbitrary element of R. Then, a = a + 1 cannot be true for all $\text{a}\in\text{A.}$ $\Rightarrow\ (\text{a, a})\notin\text{R}$ So, R is not reflexive on A. Symmetry: Let $(\text{a, b})\in\text{R}$ ⇒ b = a + 1 ⇒ -a = -b + 1 ⇒ a = b - 1 Thus, $(\text{b, a})\notin\text{R}$ So, R is not symmetric on A.Transitivity: Let (1, 2) and (2, 3) $\in\text{R}$
⇒ 2 = 1 + 1 and 3 = 2 + 1 is true. But $3\neq1+1$ $\Rightarrow\ (1,3)\notin\text{R}$ So, R is not transitive on A.

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