   Chapter 1.3, Problem 4E

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# Give an example of mappings f and g such that one of f or g is not onto but f ∘ g is onto.

To determine

An example of mapping f and g such that one of f or g is not onto but fg is onto.

Explanation

Formula used:

1) A standard way to demonstrate that f:AB is onto is to take an arbitrary element b in B and show that there exists an element aA such that b=f(x).

2) To show that a given mapping f:AB is not onto, find single element b in B for which no xA exists such that b=f(x).

3) Definition: Let g:AB and f:BC. The composite mapping fg is the mapping from A to C defined by (fg)(x)=f(g(x)) for all xA.

Explanation:

Let f:{1,1} defined by f(x)=(1)x

Also g: defined by g(x)=3x

By using definition of a composition function,

fg:{1,1} is defined by

(fg)(x)=f(g(x))=f(3x)=(1)3x

Since (fg)(2)=(1)32=(1)6=1 and

(fg)(1)=(1)31=(1)3=1

This implies 1 has preimage 1 and 1 has a preimage 2 in

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