Question

$R$ is a relation on the set $Z$ of integers and it is given by $(x,y)\in R\iff |x-y|\le 1$. Then, $R$ is

• A. reflexive and transitive
• B. reflexive and symmetric
• C. symmetric and transitive
• D. an equivalence relation

Answer 1

Answer: (B)

reflexive and symmetric

Let x be a element in Z,

then $|x-x|=0\le 1$.

So every element of Z is related to itself, Thus R is a reflexive relation .

Let x,y be two element in Z such that $|x-y|\le 1$,

then $|y-x|\le 1$.

So, $xRy\iff yRx$ and thus R is a symmetric relation .

Now let's prove that R is not transitive by an example to contradict,

$(2,1)\Rightarrow |2-1|\le 1$ is in $R$ and $(1,0)\Rightarrow |1-0|\le 1$ is also in $R$ but $(2,0)\Rightarrow |2-0|\ge 1$ is not in $R$.

Thus B is correct answer .