Let Z be the set of integers and let X 1 1 Define a functio

Let Z be the set of integers and let X = {1, -1}. Define a function

h: Z -> X by the rule   h(w) = 1, if w is even

h(w) = -1, if w is odd.

Answer the following questions:

(a1) Is h one-to-one?                    

(a2) Give your rationale for your answer in (a1). In particular, if your answer is No, give a specific counterexample.   

(b1) Is h onto?                             

(b2) Give your rationale for your answer in (b1). In particular, if your answer is No, give a specific counterexample.

Solution

(a1)

No, h is not one-one function.

(a2).

Take w = 2 and w = 4

so we get h(2) = h(4) = 1

but 2 is not equal to 4.

so for two different values of w we get same value.so not one-one.

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(b1)

Yes, h is onto.

(b2)

here codomain = {1 , -1}

Range = {1,-1}

so codomain = Range

hence onto function.