Determine whether the following relation is reflexive,symmetric,and transitive:
Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x - y \text{ is an integer}\}$
Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x - y \text{ is an integer}\}$
Given $R = \{(x, y) : x - y \text{ is an integer}\}$ for $x, y \in Z$.
$1$. Reflexive: For any $x \in Z$,$x - x = 0$,which is an integer. Thus,$(x, x) \in R$. Therefore,$R$ is reflexive.
$2$. Symmetric: Let $(x, y) \in R$. Then $x - y$ is an integer. This implies $-(x - y) = y - x$ is also an integer. Thus,$(y, x) \in R$. Therefore,$R$ is symmetric.
$3$. Transitive: Let $(x, y) \in R$ and $(y, z) \in R$. Then $x - y$ and $y - z$ are integers. Their sum $(x - y) + (y - z) = x - z$ is also an integer. Thus,$(x, z) \in R$. Therefore,$R$ is transitive.
Conclusion: $R$ is reflexive,symmetric,and transitive.
$1$. Reflexive: For any $x \in Z$,$x - x = 0$,which is an integer. Thus,$(x, x) \in R$. Therefore,$R$ is reflexive.
$2$. Symmetric: Let $(x, y) \in R$. Then $x - y$ is an integer. This implies $-(x - y) = y - x$ is also an integer. Thus,$(y, x) \in R$. Therefore,$R$ is symmetric.
$3$. Transitive: Let $(x, y) \in R$ and $(y, z) \in R$. Then $x - y$ and $y - z$ are integers. Their sum $(x - y) + (y - z) = x - z$ is also an integer. Thus,$(x, z) \in R$. Therefore,$R$ is transitive.
Conclusion: $R$ is reflexive,symmetric,and transitive.
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