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For equivalence it should satisfy,Reflexive,Transitive,Symmetric.Lets check:-

1- these relations on the set of all functions from Z to Z. Determine the properties they lack for equivalence relation?

a) {(f, g) | f (0) = g(0) or f (1) = g(1)}

 Reflexive: f(0)=f(0), so reflexive

Symmetric:- If  f (0) = g(0) or f (1) = g(1) => g(0) =f(0) or  g (1) = f(1) : so symmetric

Transitive :-  f (0) = g(0) and g(1) = g'(1)  => f (0) = g'(0) or f (1) = g('1) ,False,it is not necessary

Hence not ransitive and not equivalence

b) {(f, g) | f (0) = g(1) and f (1) = g(0)}

Reflexive :f(0)!=f(1) , hence not reflexive and not  equivalenve

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