A) Reflexive
B) Symmetric
C) Transitive
D) An equivalence relation
Correct Answer: D
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\[(x,y)\in \operatorname{R}.\], then (x -y) is an integer \[\lambda \] (say) and also \[y-x=-\lambda .\]\[(\because \lambda \in \operatorname{Z}\Rightarrow -\lambda \in \operatorname{Z})\] |
| \[\therefore \]\[y-x\]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. |
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