I have these three questions they are attached and need answers 1. Write down th

ID: 2962457 • Letter: I

Question

I have these three questions they are attached and need answers

1. Write down the definition of a function in terms of a relation. Then define what it means for a function to be injective, surjective or bijective 2. Give two examples of functions from Z to Z which are: (i.) neither injective, nor surjective, (ii.) injective but not surjective, (iii.) surjective but not injective, (iv.) bijective 3. Suppose X and Y are sets and X k and Y l. How many functions are there from X to Y? How many of these are injective? How many are surjective How many are bijective? Be sure to explain your answers.

Explanation / Answer

(1) a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output.A function is called injective (or one-to-one, or an injection) if f(a) ? f(b) for any two different elements a and b of the domain. It is called surjective (oronto) if f(X) = Y. That is, it is surjective if for every element y in the codomain there is an x in the domain such that f(x) = y. Finally f is called bijective if it is both injective and surjective.

(2)

Injective not surjective: x?2x

Surjective not injective

x??x/2?

Neither: x?x2

surjective and injective: x->x+1

(3)given IXI=k,IYI=t

no of injective functions=k(k-1)

no of sujective functions=t^k-t

x?x+1

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