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Functions : Onto and One-to-one, Bijections and Function Composition 'f o g'

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Please help with the following problems on graphs and functions. Provide step by step calculations.

1. Assuming A,B not equal to no solution, define m1:AxB->A and m2:AxB-> as follows: m1(x,y)=x and m2(x,y)=y. If f:
A->B, show that
a) f onto=>m2 |f is onto
b)f one-to-one=>m2 f is one-to-one

2. Assuming f: A->B and g: B->C are bijections, show that (g o f)^-1 = f^-1 o g^-1

3. Find a bijection from A to B when
a) A=[-2,3] and B=[5,14]
b) A=(0,infinity)and B=(-infinity, infinity)
c) A=Natural numbers and B={3,6,9,12,...}

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Solution Summary

This posting involves a problem about graphs and functions. Onto and One-to-one, Bijections and Function Composition are investigated. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question. Step by step calculations are given for each problem.

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1. Proof:
(a) f is onto, then for any y in B, we can find x in A, such that f(x)=y. Then we have
m2(x,f(x))=f(x)=y. So m2 f is onto.
(b) f is one-to-one, then if ...

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