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  3. prove the following statements! 1. let s = {0, 1,

Prove the following statements! 1. Let S = {0, 1, . . . , 23} and...

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Advance Math

Prove the following statements!

1. Let S = {0, 1, . . . , 23} and define f : Z→S by f(k) = rwhen 24|(k−r). If g : S→S is defined by

(a) g(m) = f(7m) then g is injective and

(b) g(m) = f(15m) then g is not injective.

2. Let f : A→B and g : B→C be injective. Then g ◦f : A→C isinjective.

3. Let f : A→B and g : B→C be surjective. Then g ◦ f : A→C issurjective.

4. There is a surjection f : A→B such that f −1 : B→A is not afunction.

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