4.19:

Let f : A → B and G : B → C be functions. Let h : A → C be their
composition.

(a) Prove that if f and g are surjections, then h its a
surjection.

(b) Prove that is f and g are injections, then h is a injection

4.24:

The diagram below represents a function.

(a) Prove that if g ◦ f is a total bijective function, then f is a
total injection.

(b) Prove that if g ◦ f is a total bijective function, then g is a
surjection.

4.34:

Suppose f : A → B is a total function, and A is finite. Replace *
in each of the following statements with ≤, =, ≥ to produce the
strongest correct version of the following statements.

(a) |f(A)|∗|B|

(b) If f is a surjection, then |A|*|B|

(c) If f is a surjection, then |f(A)|*|B|

(d) If f is an injection, then |f(a)|*|A|

(e) If f is a total bijection, then |A|*|B|

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