R is a relation on the set Z of integers and it is given by (x, y) ∈ R ⇔ | x − y | $\leq$ 1. Then, R is ______________ .

Question

KollegeApply · Accepted Answer

Reflexive and symmetric Reflexivity: Let x∈R. Then, x−x=0 < 1 $\Rightarrow$ |x−x | $\leq$ 1 $\Rightarrow$(x, x) ∈ R for all x ∈ Z So, R is reflexive on Z. Symmetry : Let (x, y)∈R. Then, | x−y | $\leq$ 0 $\Rightarrow$ |−(y−x) | $\leq$ 1 $\Rightarrow$ | y−x | $\leq$ 1 [ Since |x−y|=|y−x| ] $\Rightarrow$ (y, x) ∈ R for all x, y ∈ Z So, R is symmetric on Z. Transitivity : Let (x, y) ∈ R and ( y, z ) ∈ R. Then, | x−y | $\leq$ 1 and | y−z | $\leq$ 1 $\Rightarrow$ It is not always true that |x−y| $\leq$ 1. $\Rightarrow$ (x, z) ∉ R So, R is not transitive on Z.

AChoose Your Answer