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Unformatted text preview: ) ∈ A × C : ∃ b ∈ B, s.t. ( a,b ) ∈ f and ( b,c ) ∈ g } . Theorem 4.4. Let f : A → B and g : B → C , then ( a ) if f ang g are injective, then g ◦ f is injective ( b ) if f ang g are surjective, then g ◦ f is surjective ( c ) if f ang g are bijective, then g ◦ f is bijective. Deﬁnition 4.5. Let f : A → B . The inverse of f is the relation f1 = { ( y,x ) ∈ B × A : ( x,y ) ∈ f } . Theorem 4.6. Let f : A → B be a bijection, then f1 is a function, and ( a ) f1 : B → A is bijective ( b ) f1 ◦ f = i A , f ◦ f1 = i B (the identity maps). Theorem 4.7. Let f : A → B and g : B → C be bijective, then the composition g ◦ f is bijective, and ( g ◦ f )1 = f1 ◦ g1 ....
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.
 Spring '08
 Akhmedov,A
 Sets

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