Composition of Functions

# Composition of Functions - Math 100 Composition of...

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Unformatted text preview: Math 100 Composition of Functions Martin H. Weissman 1. A Composition Exercise Let A = { a,b,c,d } . Let B = { 1 , 2 , 3 } . Define functions f : A → B , and g : B → A , by the following: • f ( a ) = 1. f ( b ) = 3. f ( c ) = 3. f ( d ) = 2. • g (1) = a . g (2) = d . g (3) = c . Draw arrow diagrams for the composite function f ◦ g : B → B . Draw arrow diagrams for the composite function g ◦ f : A → A . Are the functions f and g injective? Surjective? What about the composite functions f ◦ g and g ◦ f ? 1 2. Properties of Composition Whenever A is a set, there is a special function from A to A , called the identity function . We use the notation Id A : A → A , for the identity function. It is defined by: Id A ( a ) = a, for all a ∈ A. It is important to understand what equality means for functions. Definition 1. Suppose that A and B are sets. Suppose that f and g are both functions from A to B . We say that f = g , if for all a ∈ A , f ( a ) = g ( a ). Theorem 2 (Identity Functions) . Suppose that A,B are sets, and f : A → B is a function. Then f ◦ Id A = f , and Id B ◦ f = f . Proof. If a ∈ A , then [ f ◦ Id A ]( a ) = f ( Id A ( a )) = f ( a ). Hence f ◦ Id A = f . If a ∈ A , then [ Id B ◦ f ]( a ) = Id B ( f ( a )) = f ( a ). Hence Id B ◦ f = f . / Composition of functions is an associative operation , in the following sense:...
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Composition of Functions - Math 100 Composition of...

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