Well, for (a) use the definition of "equivalence relation"!

a) (reflexive). For every x in A xEx. If x in A is f(x)= f(x)?

b) (symmetric). For all x, y in A if xEy then yEx. If x and y are in A and f(x)= f(y) is f(y)= f(x)?

c) (transitive). For all x, y, z, in A, if xEy and yEz then xEz. If x, y, z are in A, f(x)= f(y) and f(y)= f(z) is f(x)= f(z)?

b) "A/E" is the set consisting of all "equivalence classes"- subsets of A such that every x, y in that subset have xEy.

Did this problem actually say "if [a]_E = [a']_E"? In that case this is trivial. By definition of "function" if f is a function and u= v then f(u)= f(v). Much more interesting would be "verify that phi([a]_E) = phi([a']_E) if a'Ea ". But that's still easy. If a'Ea then what can you say about f(a') and f(a)?

c) Not "phi dot j", phi j, the composition. Okay, for any a, j(a) is the equivalence class containing a. What is true about f(x) foreverymember, x, of that class?