Answer
True. Proof: Suppose X is any set and f, g, and h are functions from X to X such that h is one-to-one and h ◦ f =
h ◦ g. [We must show that for all x in X, f (x) = g(x).] Suppose x is any element in X. Because h ◦ f = h ◦ g, we
have that (h ◦ f )(x) = (h ◦ g)(x) by definition of equality of
functions. Then, by definition of composition of functions,
h( f (x) = h(g(x)). But since h is one-to-one, this implies
that f (x) = g(x) [as was to be shown].
Work Step by Step
True. Proof: Suppose X is any set and f, g, and h are functions from X to X such that h is one-to-one and h ◦ f =
h ◦ g. [We must show that for all x in X, f (x) = g(x).] Suppose x is any element in X. Because h ◦ f = h ◦ g, we
have that (h ◦ f )(x) = (h ◦ g)(x) by definition of equality of
functions. Then, by definition of composition of functions,
h( f (x) = h(g(x)). But since h is one-to-one, this implies
that f (x) = g(x) [as was to be shown].
