Question

The identity function on A, idA: A →A is defined by idA(a) = a.

1. (a) We defined a function as a relation with a particular property, thus
we can talk about properties of relations. Prove that if f : A →A
is reflexive, then f =idA.
(b) Prove that if f is symmetric then the function f^2 : A →A defined
by f^2(a) = f (f (a)) is the identity.
(c) Prove that if f is transitive then the function g : im(f) →im(f) given
by g(f (a)) = f (f (a)) is the identity on im(f) .

Answer 1

These are just sketches, you should polish them yourself. In analogy to relations, I will write aFb to mean the statement f(a) = b.

(a) aFa, so f(a) = a. This is the definition of the identity function.

(b) aFb => bFa, so f(a) = b and f(b) = a. Therefore f(f(a)) = a by substitution, and hence f^2 is the identity function.

(c) aFb and bFc => aFc. So f(f(a)) = c, and f(a) = c. Thus f(c) = c, which is the identity. Make sure you sort out the im(F) stuff when you clean this up.