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ex2revsol

# ex2revsol - MAT 300 Mathematical Structures Selected...

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MAT 300, Mathematical Structures Spring 2010 Selected Solutions for Exam 2 Review We answered all questions from part (II) in class, and also parts (b), (f), and (g) from part (III). The remaining parts of (III) are answered here. III. (a) Suppose f : A B and g : B C , and assume that g f is one-to-one and onto. Prove that f is one-to-one and g is onto. Proof: Assume that f : A B and g : B C are functions and g f is one-to-one and onto. First we will prove that f is one-to-one. Let x 1 , x 2 A and assume that f ( x 1 ) = f ( x 2 ). Then, since g is a function, g ( f ( x 1 )) = g ( f ( x 2 )). But g f is one-to-one. This means that whenever g ( f ( x 1 )) = g ( f ( x 2 )), we must have x 1 = x 2 . Since x 1 = x 2 whenever f ( x 1 ) = f ( x 2 ), we conclude that f is one-to-one. Now, we wish to prove that g is onto. Let z C . Since g f is onto, there exists some x A for which g ( f ( x )) = z . Let y = f ( x ) B . Then z = g ( f ( x )) = g ( y ) for y = f ( x ) B . This shows that g is onto.

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ex2revsol - MAT 300 Mathematical Structures Selected...

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