Question

# Test whether the following relation is (1) reflexive (2) symmetric and (3) transitive R on Z defined by (a,b) $\in R\Leftrightarrow \left| a-b \right|\le 5.$

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Hint: we will have to know about the term reflexive, symmetric and transitive so that we can understand the question. For a relation R in set A. The relation said to be reflexive if (a,a) $\in$R for every a $\in$A, for symmetric relation if (a,b) $\in$R then (b,a) $\in$R and for transitive relation if (a,b) $\in$R, (b,c) $\in$R then (a,c) $\in$R.

Given the relation is R on Z defined by (a,b) $\in R\Leftrightarrow \left| a-b \right|\le 5.$
Clearly, we can say that the above relation is reflexive as $\forall a\in Z,(a,a)\in R\text{ since }\left| a-a \right|=0\le 5.$
\begin{align} & \text{Also the relation is symmetric as }\left| b-a \right|=\left| a-b \right|\le 5\text{ so (a,b)}\in R,\forall a,b\in Z. \\ & \text{But the relation is not transitive as (1,2) }\in \text{R,(2,7)}\in \text{R but (1,7)}\notin \text{R}\text{.} \\ \end{align}