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Let S = ℤ+, a ~ b if a − b is divisible by 2. Is S an equivalence relation?

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For any x \in {Z^ + } x-x=0 is divisible by 2, so, x~x and S is a reflexive relation.

If a-b is divisible by 2 then b-a=-(a-b) is divisible by 2 then a \sim b \Rightarrow b \sim a S is a symmetrical relation.

Let a~b and b~c. Then a-b is divisible by 2 and b-c is divisible by 2 .

Then the last expression can be presented in the form

a - b = 2n,\,\,n \in Z,\,\,b - c = 2k,\,\,k \in Z


b = 2k + c \Rightarrow a - b = a - 2k - c = 2n \Rightarrow a - c = 2(n + k),\,\,n + k \in Z .

Then a-c is divisible by 2 and a~c, whence S is a transitive relation.

Since S is reflexive, symmetrical and transitive relation, then S is equivalence relation.

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