   Chapter 1.5, Problem 8E

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# a. Prove that the set of all onto mappings from A to A is closed under composition of mappings.b. Prove that the set of all one-to-one mappings from A to A is closed under composition of mappings.

(a)

To determine

To prove: The set of all onto mappings from A to A is closed under composition of mappings.

Explanation

Proof:

Let f and g be two onto mappings from A to A.

f:AAg:AA

Let zA. Since f is onto, therefore there exists yA such that

f(y)=z ...... (1)

Now, since g is onto, therefore there exists xA such that

g(x)=y ...... (2)

Now,

fg(x)=f(g(x))

(b)

To determine

To prove: The set of all one-to-one mappings from A to A is closed under composition of mappings.

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