If is a relation on the set of integers given by such that for some integer . Then, is
an equivalence relation
Explanation for the correct option:
Step 1: Check for reflexive relation.
A relation is reflexive if exists.
A relation is symmetric, if both and exists.
A relation is transitive, if and exists implies that also exists.
Given such that for some integer .
In a reflexive relation, exists, substitute , which gives
On solving
Thus, a value of exists,
Therefore, the given relation is reflexive.
Step 2: Check for symmetric relation.
Assume exists which implies and also exists which implies .
Let
where,
here, is an integer, then is also an integer.
Therefore, the relation is symmetric.
Step 3: Check for transitive relation.
Assume exists which implies and also exists which implies .
Let and exists where are integers.
Multiply both the equation
Here are integers, then its sum is also an integer.
Therefore, also exist that is also exists.
Hence, the relation is transitive.
If a relation is reflexive, symmetric, and transitive, it is said to be an equivalence relation.
Hence, the correct option is (A).