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Unless otherwise noted, all sets in this module are finite. Prove the following statements... 1. Let S...

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Advance Math

Unless otherwise noted, all sets in this module are finite.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 isinjective 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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