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
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.
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.
(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|