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Problem 4.19. Let f : A → B and g : B → C be functions. Let h:A Cbe their composition. (a) Prove that if f and g are surjections, then h is a surjection (b) Prove that is f and g are injections, then h is an injection 4.24 The diagram below represents a function. (a) Prove that if go f is a total bijective function, then f is a total injection (b) Prove that if go f is a total bijective function, then g is a surjection Problem 4.34 Suppose f: A -B is a total function, and A is finite. Replace * in each of the following statements with S,,2 to produce the strongest correct version of the following statements (a) (A) B (b) If f is a surjection, then |A IBI (c) If f is a surjection, then If(A) BI (d) If f is an injection, then |f(A)A (e) If f is a total bijection, then |AB

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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