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Consider the relation on the set of non-negative integers defined by $x \equiv y$ if and only if:

  1. $x$ $\text{mod}$ $3=3$ $\text{mod}$ $y$
  2. $3$ $\text{mod}$ $x \equiv 3$ $\text{mod}$ $y$
  3. $x$ $\text{mod}$ $3=y$ $\text{mod}$ $3$
  4. None of the above
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Relation R = {(x,y) | x mod 3 = y mod 3}

For Example

4 mod 3 =1

7 mod 3 = 1

so (4,7) ∈ R

Reflexive : x mod 3 = x mod 3

Symmetric : If x mod 3 = y mod 3 then y mod 3 = x mod 3

Transitive : If x mod 3 = y mod 3, and y mod 3 = z mod 3, then x mod 3 = z mod 3

Therefore, the relation R is equivalence relation

 

 

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