midterm2sols

midterm2sols - Introduction to Advanced Mathematics Midterm...

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Introduction to Advanced Mathematics — Midterm II November 3rd, 2011 Write your answers in the space provided after each question. Please make an effort to write neatly. The problems marked by refer directly to material in the class notes or your homework. You may want to start with those questions—they make up more than half of the test. Your Name: KEY 1
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Let f : A B be a function. Give the definition of left-inverse of f , and the definition of right-inverse of f . –A function g : B A is a left-inverse of f if g f = 1 A . –A function g : B A is a right-inverse of f if f g = 1 B . (Class notes, Definition 12.1.) Let f : A B be a function. Prove that if f has a right-inverse, then f is surjective. (We have also proved that if f is surjective, then it has a right-inverse. You don’t need to prove this now.) Let g : B A be a right-inverse of f : therefore, f g = 1 B . Let b B be any element. We have to show that there is an element a A such that f ( a ) = b . I claim that g ( b ) is such an element. Indeed, f ( a ) = f ( g ( b )) = f g ( b ) = 1 B ( b ) = b as needed. (This is the argument given in the proof of Lemma 13.2, for the particular case g f = 1 A .) Let f : A B be a function. Prove that f is injective if and only if for all subsets S A , S = f - 1 ( f ( S )) . First assume that S = f - 1 ( f ( S )) for all subsets S A . In order to prove that f is injective, we have to show that if f ( a ) = f ( a 0 ) , then a = a 0 . Let S = { a } . Since f ( a 0 ) = f ( a ) , we have a 0 f - 1 ( f ( S )) . Since f - 1 ( f ( S )) = S = { a } , this says that a 0 ∈ { a } , and it follows that a 0 = a , as needed. Next, assume that
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midterm2sols - Introduction to Advanced Mathematics Midterm...

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