In the set `Z` of all integers, which of the following relation `R` is not an equivalence relation? `x\ R\ y :` if `xlt=y` (b) `x\ R\ y :` if `x=y` (c) `x\ R\ y` : if `x-y` is an even integer (d) `x\ R\ y` : if `x=y` (mod 3)

Answer

c) xRy : if `x-y` is an even integer

Explanation

For option C, x Ry:if x-y is an even integer.

Let a be any number in {Z},

Then,` a-a=0 `is an even integer .

So, Every number is related to itself.

Thus the relation is reflexive.

Let a, b be two numbers in Z,

Also let `a-b=k` be an even integer (which implies a R b )

Then, `b-a=-k` is also an even integer (which implies b R a ).

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