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

Last updated date: 26th Feb 2024
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IVSAT 2024
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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.

Complete step-by-step answer:
Given the relation is R on Z defined by (a,b) \[\in R\Leftrightarrow \left| a-b \right|\le 5.\]
Now, we will check the condition of reflexive, symmetric and transitive for the above relation.

Clearly, we can say that the above relation is reflexive as \[\forall a\in Z,(a,a)\in R\text{ since }\left| a-a \right|=0\le 5.\]
 & \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{.} \\

Therefore, the above given relation is reflexive, symmetric but not transitive.

NOTE: Just remember the term reflexive, symmetric and transitive so you can easily understand the given question and solve it easily. The condition for the relation to be reflexive, symmetric and transitive are mentioned in the hint.
Also, remember the point that if any relation is symmetric,reflexive as well as transitive then the relation is known as equivalence relation.In many questions we have to find the equivalence relation also so it is very important to remember this point.