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Q.
Let $R$ be the relation in the set $Z$ of all integers defined by $R=\{(x, y): x-y$ is an integer $\}$. Then $R$ is
Relations and Functions - Part 2
Solution:
Here, $R=\{(x, y): x-y$ is an integer $\}$ is a relation in the set of integers.
For reflexivity, put $y=x, x-x=0$ which is an integer for all $x \in Z$. So, $R$ is reflexive in $Z$.
For symmetry, let $(x, y) \in R$, then $(x-y)$ is an integer $\lambda$ (say) and also $y-x=-\lambda$. $(\because \lambda \in Z \Rightarrow-\lambda \in Z)$ $\therefore y-x$ is an integer $\Rightarrow(y, x) \in R$. So, $R$ is symmetric.
For transitivity, let $(x, y) \in R$ and $(y, z) \in R$, so $x-y=$ integer and $y-z=$ integers, then $x-z$ is also an integer.
$\therefore (x, z) \in R$. So, $R$ is transitive.