If is the set of integers. Then, the relation on is
reflexive and symmetric but not transitive
Explanation for the correct option.
Step 1. Check whether the relation is reflexive.
A relation in is reflexive if every ordered pair where is in .
It is given that relation is defined as: .
Now, the ordered pair is in relation if , so is always true
As square of an integer is always non-negative so is always true.
Hence, for every there is an ordered pair in relation . So the relation is reflexive.
Step 2. Check whether the relation is symmetric.
A relation is symmetric if for every ordered pair in the relation there is also an ordered pair in it.
If is in ordered pair satisfying the relation then .
Now it is known that multiplication is commutative so .
And so if is true then is also true.
And thus is also in the relation .
So there is an ordered pair in for every in . So the relation is symmetric.
Step 3. Check whether the relation is transitive.
A relation is transitive if for every there is an in .
The ordered pair is in relation because is true.
The ordered pair is in relation because is true.
Now,
As is a false statement so the inequality is false and so the ordered pair does not belong to relation .
So relation contains and but not . So relation is not transitive.
Thus, the relation defined as is reflexive and symmetric but not transitive.
Hence, the correct option is C.