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Q.
The relation $R$ defined in the set $A=\{1,2,3,4,5$, $6,7\}$ by $R=\{(a, b)$ : both $a$ and $b$ are either odd or even $\}$. Then, $R$ is
Relations and Functions - Part 2
Solution:
Given, any element a in $A$, both a and a must be either odd or even, so that $(a, a) \in R$. Further, $(a, b) \in R \Rightarrow$ both a and $b$ must be either odd or even $\Rightarrow(b, a) \in R$. Similarly, $(a, b)$ $\in R$ and $(b, c) \in R \Rightarrow$ all elements $a, b, c$ must be either even or odd simultaneously $\Rightarrow(a, c) \in R$. Hence, $R$ is an equivalence relation.