   Chapter 1.3, Problem 7E

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# a. Give an example of mappings f and g , where f is onto, g is one-to-one, and f ∘ g is not one-to-one.b. Give an example of mappings f and g , different from example 3 , where f is onto, g is one-to-one, and f ∘ g is not onto.

(a)

To determine

The function f and the function g such that the function f is one-to-one, the function g is onto and the composite function fg is not one-to-one.

Explanation

Assume the function g(x)=5x, xZ and f(x)={x/5ifxisdivisibleby53ifxisnotdivisibleby5, xZ.

Here, set of integers is Z.

Let m,nZ such that mn, then,

g(m)g(n)

Thus, g is one-to-one.

For x=2, g(2)=3 and for x=20, f(20)=4.

Thus, f is onto.

Calculate fg(m)

fg(m)=f(g(m))=f({m/5

(b)

To determine

The function f and the function g such that the function f is onto, the function g is one-to-one but the composite function fg is not onto.

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