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Functions

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**Definition and notation**

Definition: A function f from a set X to a set Y is a relationship between elements of X and Y with the property that each element of X is related to a unique element of Y. Denoted f:X→Y . X is called domain of f; Y is called co-domain of f. Example: X=Z, Y=Z and f : x → 2∙x / 2 xX ! yY such that f(x)=y . f(x) is called f of x (or image of x under f). Range of f = {yY | y=f(x) for some x in X} Inverse image of y = {xX | f(x)=y} Example(cont.): range of f = all even integers ; inverse image of 4 = {3, 4} .

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**Examples of Functions Squaring function: f : x → x2 .**

Constant function: f : x → 3 . Linear function: f : x → 3x+2 . Factorial function: f : n → n! . Any sequence can be considered as a function defined on a set of integers. E.g., sequence 2,5,8,11,14,… is a function from Z+ to Z+ defined as follows f : n → 3n-1

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**Boolean Functions Recall the truth tables:**

Can be considered as a function; the domain is the set of all ordered couples of 0 and 1; the co-domain is {0,1} . Input Output p q p q 1

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**Boolean Functions Definition:**

An (n-place) Boolean function is a function whose domain is the set of all ordered n-tuples of 0’s and 1’s and whose co-domain is the set {0,1}. Example: f : (x,y,z) → (~x y) z

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**One-to-one Functions Definition:**

Let F be a function from set X to set Y. F is one-to-one (or injective) iff for all elements x1, x2 X if F(x1)=F(x2) then x1=x2 . Examples: Define f : Z → Z by f(n)=2n+3 ; g : R → R by f(x)=x2 . Then f is one-to-one, and g is not.

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**Onto Functions Definition: Let F be a function from set X to set Y.**

F is onto (or surjective) iff for any element y Y there is a x X such that F(x)=y . Examples: Define f : Z → Z by f(n)=2n+3 ; g : Z → Z by f(n)=n-2 . Then g is onto, and f is not.

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**Exponential Functions**

The exponential function with base b is the following function from R to R+ : expb(x) = bx b0=1 b-x = 1/bx bubv = bu+v (bu)v = buv (bc)u = bucu

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**Logarithmic Functions**

The logarithmic function with base b (b>0, b1) is the following function from R+ to R: logb(x) = the exponent to which b must raised to obtain x . Symbolically, logbx = y by = x . Properties:

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**One-to-one Correspondences**

Definition: A one-to-one correspondence (or bijection) from a set X to a set Y is a function f:X→Y that is both one-to-one and onto. Examples: 1) Linear functions: f(x)=ax+b when a0 (with domain and co-domain R) 2) Exponential functions: f(x)=bx (b>0, b1) (with domain R and co-domain R+) 3) Logarithmic functions: f(x)=logbx (b>0, b1) (with domain R+ and co-domain R)

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**Inverse Functions Theorem:**

Suppose F: X→Y is a one-to-one correspondence. Then there is a function F-1: Y→X defined as follows: Given any element in Y, F-1(y) = the unique element x in X such that F(x)=y . The function F-1 is called the inverse function for F. Example: The logarithmic function with base b (b>0, b 1) is the inverse of the exponential function with base b.

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Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.

Fall 2015 COMP 2300 Discrete Structures for Computation Donghyun (David) Kim Department of Mathematics and Physics North Carolina Central University 1.

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