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Sec 2.3 Functions

Definition 1

Let A and B be nonempty sets. A function f from A to B is anassignment of exactly one element of B to each element of A. We write f  (a) = b if b is the unique element of B assigned by the function f to the element a of A. If f is a function from A to B, we write f  :  A → B .

Remark:  Functions are sometimes also called  mappings ortransformations.

A function  f  :  A → B  can also be defined in terms of a relation fromA  to  B  which is just a subset of  A

×B .

A function  f  :  A → B   is defined in terms of a relation from  A  to  B needs to contains one, and only one, ordered pair (a, b ) for everyelement  a ∈ A.

See Figure 2 Page 134.

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Sec 2.3 Functions

Definition 2

If f is a function from A to B, we say that A is the  domain of f and B is the  codomain of f . If f  (a) = b, we say that b is  the  image  of a and a is apreimage  of b. The  range  of f is the set of all images of elements of A.Also, if f is a function from A to B, we say that f   maps  A to B.

Two functions are  equal when they have the same domain, have thesame codomain, and map element of their common domain to thesame elements in their common codomain.

•Exercise 2 page 146: Determine whether  f   is a function from  Z to  R  if.a)f  (n) =

 ±n   b)f  (n) =

√ n2 + 1 c)f  (n) = 1/(n2

−4).

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Sec 2.3 Functions

•Exercise 6 page 146: Find the domain and range of these functions.a)  the function that assigns to each pair of positive integers the first

integer of the pairb)  the function that assigns to each positive integer its largest decimal

digitc)  the function that assigns to a bit sting the number of ones minus the

number of zeros in the stringd)  the function that assigns to each positive integer the largest integernot exceeding the square root of the integer.

e)  the function that assigns to a bit string the longest stings of ones inthe strings.

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Sec 2.3 Functions

Definition 3

Let f 1  and f 2  be functions from A to  R. Then f 1 + f 2  and f 1f 2  are also functions from A to  R  defined by •  (f 1 + f 2)(x ) = f 1(x ) + f 2(x ),   •  f 1f 2(x ) = f 1(x )f 2(x ).

Definition 4

Let f be a function from the set A to the set B and let S be a subset of A.The  image  of S under the function f is the subset of B that consists of the images of the elements of S. We denote the image of the S by f  (S ), so 

f  (S ) =

 {t 

 | ∃s 

 ∈ S (t  = f  (s ))

}.

We also use the shorthand  {f  (s ) | s  ∈ S }  to denote this set.Remark:f  (S ) denotes a set, and not the value of the function  f   for theset  S .

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Sec 2.3 Functions: One-to-One and Onto Functions

Definition 5

A function f is said to be  one-to-one , or   injective , if and only if f  (a) = f  (b )  implies that a = b for all a and b in the domain of f . Afunction is said to be an injection if it is  one-to-one .

Contrapositive of the implication in the definition:   A function  f 

is  one-to-one  if and only if  f  (a) = f  (b ) whenever  a = b .Remark:  Expressing  f   is  one-to-one  using quantifiers as∀a∀b (f  (a) = f  (b ) → a  =  b ) or equivalently∀a∀b (a = b  → f  (a) = f  (b )), where the universe of discourse is thedomain of the function.

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Sec 2.3 Functions

•Exercise 10 page 146: Determine whether each of these function from

{a, b , c , d 

} to itself is one-to-one.

a)   f  (a) = b , f  (b ) = a, f  (c ) = c , f  (d ) = d .b)   f  (a) = b , f  (b ) = b , f  (c ) = d , f  (d ) = c c)   f  (a) = d , f  (b ) = b , f  (c ) = c , f  (d ) = d 

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Sec 2.3 Functions: Some conditions that guarantee that a

function is one-to-one

Definition 6

A function f whose domain and codomain are subsets of the set of real numbers is called increasing if f  (x ) ≤ f  (y ), and strictly increasing if f  (x ) <  f  (y ), whenever x    f  (y ), whenever x  

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Sec 2.3 Functions

•Exercise 12 page 146: Determine whether each of these functions fromZ  to  Z  is one-to-one.

a)f  (n) = n − 1 b)f  (n) = n2 + 1 c)f  (n) = n3•Example 1  What are the truth values of these propositions? Justify youranswers.

A function that is increasing but not strictly increasing is necessary

one-to-one. [ ]A function that is strictly increasing or is strictly decreasing must beone-to-one. [ ]A function that is decreasing but not strictly decreasing is notnecessary one-to-one. [ ]

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Sec 2.3 Onto Functions

Definition 7A function f from A to B is called onto or surjective, if and only if for every element b  ∈ B there is an element a ∈ A with f  (a) = b. A functionf is called a surjection if it is onto.

•Remark:   A function  f   is onto if  ∀y ∃x   (f  (x ) = y ), where the domain of x   is  . . .  and the domain of  y   is   . . . .

•Question: What is the relation between the codomain and therange of the onto function?

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Sec 2.3 Functions

•  Exercise 14 pages 146-147: Determine whether  f  :  Z × Z → Z is ontoif 

a)f  (m, n) = 2m − n   b)f  (m, n) = m2 − n2 c)f  (m, n) = m + n − 1d)f  (m, n) = |m| − |n|   e)f  (m, n) = m2 − 4

Definition 8

The function f is a one-to-one correspondence, or a bijection, if it is bothone-to-one and onto.

Question: Give an example of a bijective function.Question: is the identity function ιA  on a set  A,

ιA :  A → A  where   ιA(x ) = x   for all  x  ∈ A,a bijection?

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Sec 2.3 Functions

•  Exercise 18 page 147: Determine whether each of these functions is abijection from  R to R.

a)f  (x ) = −3x  + 4 b)f  (x ) = −3x 2 + 7 c)f  (x ) = (x  + 1)/(x  + 2)d)f  (x ) = x 5 + 1.

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Sec 2.3 Functions: Inverse Functions and Compositions of 

Functions

Definition 9

Let f be a one-to-one correspondence from the set A to the set B . The inverse function of f is the function that assigns to an element b belonging to B the unique element a in A such that f  (a) = b. The inverse functionof f is denoted by f −1. Hence, f −1(b ) = a when f  (a) = b.

A one-to-one correspondence is called   invertible  because  . . . . . . .

A function is  not invertible   if it is  . . . . . . .

•See Remark Page 139 and Figure 6:

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Sec 2.3 Functions

Definition 10

Let g be a function from the set A to the set B and let f be a functionfrom the set B to the set C. The  composition of the functions f and g ,denoted by f  ◦ g, is defined by  (f  ◦ g )(x ) = f  (g (x )).•Exercise 32 page 147: Find  f  ◦ g   and  g  ◦ f  , where  f  (x ) = x 2 + 1 and

g (x ) = x  + 2, are functions from  R to R.

•See Remark Page 141 and Figure 7:

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Sec 2.3 Functions: The Graphs of Functions

Definition 11

Let f be a function from the set A to the set B . The  graph of the functionf is the set of ordered pairs  {(a, b ) | a ∈ A and f  (a) = b }.

•Exercise 58 page 148: Draw the graph of the function  f  (n) = 1 − n2from  Z to  Z.

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Sec 2.3 Functions: Some Important Functions

Definition 12

The  floor function assigns to the real number x the largest integer that is less than or equal to x. The value of the floor function at x is denoted by x . The  ceiling function assigns to the real number x the smallest integer that is greater than or equal to x. The value of the ceiling function at x is denoted by  x .•Example 2: Draw the graph of the functions  f  (x ) = x   and  f  (x ) = x from  R to  R.

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Sec 2.3 Functions: Table 1:Properties of the Floor and

Ceiling Functons

Table 1  Useful Properties of the Floor and Ceiling Func-tions (n is an integer)

(1a)x  =  n   if and only if  n ≤ x  

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Sec 2.3 Functions

•Example 27 page 145: Prove that if  x   is a real number, then

2x 

 =

 x 

+

x  +   12

.

•Exercise 60 page 148: Draw the graph of the function  f  (x ) = x /2from  R to  R.

•Exercise 69 Pages 148-149: Prove or disprove each of these statement:

b)2x  = 2x   whenever  x   is a real number.d)xy  = x y   for all real numbers  x   and  y .

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Sec 2.3 Functions

•Example 29 Page 145: The factorial function  f  :  N → Z+, denoted byf  (n) = n!.

•Exercise 16 page 147: Give an example of function from N  to N  that isone-to-one but not onto.onto but not one-to-one.both one-to-one and onto (but different from the identity function).

neither one-to-one nor onto.

Exercise 36 page 147: Let  f  be a function from the set  A  to the set  B .Let  S   and  T  be subsets of  A. Show that

a)f  (S  ∪ T ) = f  (S ) ∪ f  (T ).b)f  (S  ∩ T ) ⊆ f  (S ) ∩ f  (T )

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