Discrete Mathematics - Lecture 2.3 Functions PDF

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Discrete Mathematics - Lecture 2.3 Functions...

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Math 3336 Section 2.3 Functions • • •

Definition of a Function. Injection, Surjection, Bijection Inverse Function

Definitio assigns e We write of ฀฀.

• • •

Function Composition Graphing Functions Floor, Ceiling, Factorial

o a set ฀฀ , denoted element of ฀฀ .

rule that

ement of ฀฀ assigned by the function f to the element ฀฀

Example

Definition: Let ฀฀ and ฀฀ be sets. of (฀฀, ฀฀), where

of ฀฀ and ฀฀ , denoted by

, is the set

Given a function f : A → B:

   

We say f A is calle B is calle If f(a) =

 The rang  Two func map each

d

T

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Definition: A function f is said to be if and only if implies that for all ฀฀ and ฀฀ in the domain of ฀฀. A function is said t is one-to-one. Example:

Definition: A function ฀฀ from ฀฀ to ฀฀ there is an element ฀฀ ∈ ฀฀ with Example:

Definition: A function f is a and onto (surjective and injective).

, if it is both one-to-one

Example:

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if it

Example: Let ฀฀ be the function from {฀฀, ฀฀, ฀฀ , ฀฀} to ฀฀(฀฀ ) = and ฀฀(฀฀ ) = 3. Is f an onto function?

} defined by ฀฀(฀฀) =

Definition: Let ฀฀ be a rom ฀฀ to ฀฀. Then the from ฀฀ to ฀฀ defined as ฀฀ −1 (฀฀) = ฀฀ iff ฀฀(฀฀

of ฀฀ , denoted ฀฀

No inverse exists unless f is a bijection. W

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฀฀(฀฀ ) =

, is the function

Example: Let f be the function from {a,b,c} to {1,2,3} such that f(a) = 2, f(b) = 3, and f(c) = 1. Is f invertible and if so what is its inverse?

Example: Let ฀฀: ℤ → ℤ be such that inver

Is ฀฀ invertible, and if so, what is its

E

฀฀ ฀฀ → ฀฀ ฀฀ ฀฀ → ฀฀

฀฀

฀฀

•

Th

function, deno

Example:

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is the product of the first ฀฀ positive integers.

Example: Find f(2), f(3), f(4).

Stirling’s formula ฀฀! ~√2฀฀฀฀(฀฀/฀฀ )฀฀

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