• # question_answer 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 A) Reflexive B) Symmetric C) Transitive D) An equivalence relation

 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.