midterm2.takehomeFALL06

# midterm2.takehomeFALL06 - SECOND TAKE-HOME MIDTERM, FALL...

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DUE FRIDAY 11/17 (1) [ Hint: I recommend that you start by working out the example in part (g).] (a) (4 points): Write down the deﬁnitions of injective and well-deﬁned . Deﬁnition: An equivalence relation is called trivial, if x 1 x 2 ⇐⇒ x 1 = x 2 . Let f : X Y be a map. We deﬁne the following relation on X : x 1 f x 2 def ⇐⇒ f ( x 1 ) = f ( x 2 ) . (b) (3 points) Show that f is an equivalence relation. (c) (14 points) Prove formally that f is injective if and only if f is trivial. Consider the map p : X X/ f x 7→ [ x ] . (d) (2 points) Check that p is surjective. Recall that the image of f is deﬁned as the set im( f ) := { y Y | ( x X )( f ( x ) = y ) } . Let i : im( f ) Y denote the inclusion map of im( f ) in Y . (e) (8 points) Prove that the map g : X/ f im( f ) [ x ] 7→ f ( x ) is well-deﬁned. Date

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## This note was uploaded on 01/27/2010 for the course MATH 347 taught by Professor ? during the Spring '09 term at University of Illinois at Urbana–Champaign.

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midterm2.takehomeFALL06 - SECOND TAKE-HOME MIDTERM, FALL...

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