lec07-functions

# lec07-functions - EECS 203 Winter 2007 Discrete Mathematics...

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EECS 203, Winter 2007 Discrete Mathematics Lecture 7 Functions January 25 Reading: Rosen [2.3] January 25 Functions, Page 1

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7.1 Functions Let A and B be two sets. A function from A to B is an assignment to each element a A to exactly one element f ( a ) B . If A = B , then f is called a function on A . If f is a function from A to B , we write f : A B . We call A the domain of f , and B the range . If a A is mapped to b B , we write a b , and b is called the image of a , and a is called a pre-image of b . Let S A . The image of S , denoted by f ( S ), is the set { f ( a ) : a S } . January 25 Functions, Page 2

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7.1 Functions A function f : A B is one-to-one , or injective , if for any a 1 , a 2 A , a 1 = a 2 , f ( a 1 ) = f ( a 2 ). onto , or surjective , if for each b B , there is a A with f ( a ) = b . one-to-one correspondence , or bijective , if it is both injective

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Unformatted text preview: and surjective. January 25 Functions, Page 3 7.1 Functions Let A , B , C be sets and f : A → B , g : B → C . Then the composition of f and g , written g ◦ f is the function g ◦ f : A → C such that g ◦ f ( a ) = g ( f ( a )) , ∀ a ∈ A. The inverse of a bijection f : A → B is the function f-1 : B → A such that f-1 ◦ f is the identity function on A . The graph of a function f : A → B is the set { ( a, f ( a )) : a ∈ A } . January 25 Functions, Page 4 7.2 Some special functions • Ceiling and ﬂoor. • Factorial. January 25 Functions, Page 5...
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• Winter '07
• YaoyunShi
• Discrete Mathematics Lecture

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