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Unformatted text preview: ( ⇒ ) Assume that f : X → Y is surjective. Deﬁne a function g : Y → X by sending y to any preimage x of y under f . Note that x exists because f is surjective. We claim that g is a right inverse of f , i.e., f ◦ g = I Y . We must check that f ◦ g ( y ) = I Y ( y ) for all y ∈ Y . Now g ( y ) = x , where x satisﬁes f ( x ) = y (by deﬁnition of g ). Hence f ◦ g ( y ) = f ( g ( y )) = f ( x ) = y = I Y ( y ) . ( ⇐ ) Assume there exists g : Y → X such that f ◦ g = I Y . We must show that for all y ∈ Y there exists x ∈ X such that f ( x ) = y . Let y ∈ Y , and set x = g ( y ). Then f ( x ) = f ( g ( y )) = f ◦ g ( y ) = I Y ( y ) = y, and hence x is a preimage for y . Date : March 13th, 2008. 1...
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This note was uploaded on 06/15/2009 for the course MATH 74 taught by Professor Courtney during the Spring '07 term at Berkeley.
 Spring '07
 COURTNEY
 Math, Division

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