assnmt1 - PMath 336: Introduction to Group Theory Exercise...

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PMath 336: Introduction to Group Theory Exercise set 1 May 13, 2008 Solution should be submitted by the end of the Monday lecture of the following week, either in class or in the submission box. You may not submit joint or identical works. 1. (6 points) Let f : X -→ Y be a function between two sets. Show that f is injective if and only if, for any set Z , and any two functions h,g : Z -→ X , if f h = f g then h = g (in other words, f is injective iff it can be cancelled on the left.) Likewise, show that f is surjective if and only if, for any set Z and functions h,g : Y -→ Z , if h f = g f , then h = g ( f can be cancelled on the right.) Solution: We showed in class that f is injective if and only if there is a function t : Y -→ X with t f the identity (if X is empty, the statement is easy.) Applying t to the equality fh = fg we get h = g . Conversely, if f is not injective, f ( x ) = f ( y ) for some distinct x and y . Consider Z = { 1 } , and h,g the functions mapping 1 to x,y , respectively. If f is surjective, let h and g be two functions from Y , y an element of Y . Then y = f ( x ) for some x X . Hence h ( y ) = h ( f ( x )) = g ( f ( x )) = g ( y ), so h and g agree on all elements, hence they are equal. Conversely, if y Y is not in the image of f , let Z = { 0 , 1 } , h : Y -→ Z the constant function 0, and g the function such that g ( y ) = 1 is 0 for all other values. Then hf = gf = 0, but h and g are different. 2. (5 points) Let (
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This note was uploaded on 04/27/2010 for the course PMATH 336 taught by Professor Moshekamensky during the Spring '08 term at Waterloo.

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assnmt1 - PMath 336: Introduction to Group Theory Exercise...

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