Practicing Success
R is a relation on the set Z of integers and it is given by $(x, y) ∈R⇔|x -y | ≤1$. Then, R is |
reflexive and transitive reflexive and symmetric symmetric and transitive an equivalence relation |
reflexive and symmetric |
The correct answer is Option (2) → reflexive and symmetric For any $x ∈ Z$, we have $|x-x|=0≤1$ $∴|x-x|≤1$ for all $x ∈ Z$ $⇒(x, x) ∈ R$ for all $x ∈ Z$ ⇒ R is reflexive on Z. Let $(x, y) ∈ R$. Then, $|x-y|≤1⇒|y-x|≤1⇒ (y, x) ∈ R$ Thus, $(x, y) ∈ R⇒(y, x) ∈ R$ So, R is a symmetric relation on Z. We observe that $(1, 0) ∈ R$ and $(0, -1) ∈ R$, but $(1, -1) ∉ R$. So, R is not a transitive relation on Z. |