lec4 - ) ∈ A × C : ∃ b ∈ B, s.t. ( a,b ) ∈ f and (...

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Math 117 – April 07, 2011 4 Functions A function between two sets A and B is a nonempty relation f A × B , such that if ( a,b ) f and ( a,b 0 ) f , then b = b 0 . Domain of f is dom f = { a A : b B, s.t. ( a,b ) f } Range of f is rng f = { b B : a A, s.t. ( a,b ) f } If dom f = A , then f is said to be a function from A into B , and the notation f : A B is used. We usually use the notation y = f ( x ) for ( x,y ) f , where y is called the image of x under f . Definition 4.1. A function f : A B is called: ( a ) surjective, if rng f = B ( b ) injective, if f ( a ) = f ( a 0 ) a = a 0 ( c ) bejective, if f is both surjective and injective. If f : A B is a function, and C A , then f ( C ) = { f ( x ) : x C } is called the image of C . If D B , then f - 1 ( D ) = { x A : f ( x ) D } is called the preimage of D . Theorem 4.2. Let f : A B , and C,C 1 ,C 2 A , and D,D 1 ,D 2 B , then C f - 1 [ f ( C )] and f - 1 [ f ( C )] = C if f is injective f [ f - 1 ( D )] D and f [ f - 1 ( D )] = D if f is surjective f ( C 1 C 2 ) f ( C 1 ) f ( C 2 ) f ( C 1 C 2 ) = f ( C 1 ) f ( C 2 ) if f is injective f ( C 1 C 2 ) = f ( C 1 ) f ( C 2 ) f - 1 ( D 1 D 2 ) = f - 1 ( D 1 ) f - 1 ( D 2 ) f - 1 ( D 1 D 2 ) = f - 1 ( D 1 ) f - 1 ( D 2 ) f - 1 ( B \ D ) = A \ f - 1 ( D ) Definition 4.3. If f : A B and g : B C , then the composition of f ang g is the function ( g f )( x ) = g ( f ( x )). That is ( g f ) = { ( a,c
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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 f-1 = { ( y,x ) ∈ B × A : ( x,y ) ∈ f } . Theorem 4.6. Let f : A → B be a bijection, then f-1 is a function, and ( a ) f-1 : B → A is bijective ( b ) f-1 ◦ f = i A , f ◦ f-1 = 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 = f-1 ◦ g-1 ....
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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.

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