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