 
Summary: Review Sheet 2: Solutions to Selected Problems
1.
2. Let f : A B and g : B C be functions. Prove or disprove.
(a) If f is injective, then g f is injective.
The statement is false. Let A = B = C = {1, 2}, f the identity function
and g the constant function
g(x) = 1 for x = 1, 2.
Here, f is injective, but g f isn't.
(b) If g f is injective, then g is injective.
The statement is false. Let A = {1}, B = C = {1, 2}. Let f = {(1, 1)} and
g = {(1, 1), (2, 1)}. Here g f is injective, but g isn't.
3. Let f : R  {1} R be the function defined by f(x) = x
x1 . Show that f is
injective, but not surjective.
First, we'll show f is injective. We suppose f(x) = f(y) for some x, y R{1}.
By the definition of the function f, we have
x
x  1
=
y
