Functions

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AgendaSection 1.8: Functions Domain, co-domain, range Image, pre-image One-to-one, onto, bijective, inverse Functional composition and exponentiation Ceiling “” and floor “”

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FunctionsIn high-school, functions are often identified

with the formulas that define them. EG: f (x ) = x 2

This point of view does not suffice in Discrete Math. In discrete math, functions are not necessarily defined over the real numbers.

EG: f (x ) = 1 if x is odd, and 0 if x is even.So in addition to specifying the formula one

needs to define the set of elements which are acceptable as inputs, and the set of elements into which the function outputs.

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Functions. Basic-Terms.DEF: A function f : A B is given by a

domain set A, a codomain set B, and a rule which for every element a of A, specifies a unique element f (a) in B. f (a) is called the image of a, while a is called the pre-image of f (a). The range (or image) of f is defined byf (A) = {f (a) | a A }.

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Functions. Basic-Terms.

EG: Let f : Z R be given by f (x ) = x 2

Q1: What are the domain and co-domain?

Q2: What’s the image of -3 ?Q3: What are the pre-images of 3, 4?Q4: What is the range f (Z) ?

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Functions. Basic-Terms.

f : Z R is given by f (x ) = x 2

A1: domain is Z, co-domain is RA2: image of -3 = f (-3) = 9A3: pre-images of 3: none as 3 isn’t

an integer! pre-images of 4: -2 and 2

A4: range is the set of perfect squares f (Z) = {0,1,4,9,16,25,…}

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One-to-One, Onto, Bijection. Intuitively.

Represent functions using “node and arrow” notation:One-to-One means that no clashes occur.

BAD: a clash occurred, not 1-to-1

GOOD: no clashes, is 1-to-1

Onto means that every possible output is hit BAD: 3rd output missed, not onto

GOOD: everything hit, onto

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One-to-One, Onto, Bijection. Intuitively.

Bijection means that when arrows reversed, a function results. Equivalently, that both one-to-one’ness and onto’ness occur. BAD: not 1-to-1. Reverse

over-determined:

BAD: not onto. Reverseunder-determined:

GOOD: Bijection. Reverseis a function:

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One-to-One, Onto, Bijection. Formal

Definition.DEF: A function f : A B is:

one-to-one (or injective) if different elements of A always result in different images in B. onto (or surjective) if every element in B is hit by f. I.e., f (A ) = B.a one-to-one correspondence (or a bijection, or invertible) if f is both one-to-one as well as onto. If f is invertible, its inverse f -1 : B A is well defined by taking the unique element in the pre-image of b, for each b B.

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One-to-One, Onto, Bijection. Examples.

Q: Which of the following are 1-to-1, onto, a bijection? If f is invertible, what is its inverse?

1. f : Z R is given by f (x ) = x 2

2. f : Z R is given by f (x ) = 2x3. f : R R is given by f (x ) = x 3

4. f : Z N is given by f (x ) = |x |

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One-to-One, Onto, Bijection. Examples.

1. f : Z R, f (x ) = x 2: none 1. not 1-1 clashes for -1,1 in Z 2. not onto -1,-2 missed from R

2. f : Z R, f (x ) = 2x : 1-13. f : R R, f (x ) = x 3: 1-1, onto,

bijection, inverse is f (x ) = x (1/3)

4. f : Z N, f (x ) = |x |: onto

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CompositionWhen a function f spits out elements of

the same kind that another function g eats, f and g may be composed by letting g immediately eat each output of f.

DEF: Suppose that g : A B and f : B C are functions. Then the composite f g : A C is defined by setting

f g (a) = f ( g (a) )

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Composition. Examples.

Q: Compute g f where 1. f : Z R, f (x ) = x 2

and g : R R, g (x ) = x 3

2. f : Z Z, f (x ) = x + 1and g = f -1 so g (x ) = x – 1

3. f : {people} {people},f (x ) = the father of x, and g = f

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Composition. Examples.1. f : Z R, f (x ) = x 2

and g : R R, g (x ) = x 3

f g : Z R , f g (x ) = x 6

2. f : Z Z, f (x ) = x + 1and g = f -1

f g (x ) = x (true for any function composed with its inverse)

3. f : {people} {people},f (x ) = g(x ) = the father of x

f g (x ) = grandfather of x from father’s side

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Repeated CompositionWhen the domain and codomain are equal, a

function may be self composed. The composition may be repeated as much as desired resulting in functional exponentiation. The whole process is denoted by

f n (x ) = f f f f … f (x ) where f appears n –times on the right side.

Q1: Given f : Z Z, f (x ) = x 2 find f 4

Q2: Given g : Z Z, g (x ) = x + 1 find g n

Q3: Given h(x ) = the father of x, find hn

n

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Repeated Composition

A1: f : Z Z, f (x ) = x 2. f 4(x ) = x (2*2*2*2) = x 16

A2: g : Z Z, g (x ) = x + 1 gn (x ) = x + n

A3: h (x ) = the father of x, hn (x ) = x ’s n’th patrilineal ancestor

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Ceiling and FloorThis being a course on discrete math, it is

often useful to discretize numbers, sets and functions. For this purpose the ceiling and floor functions come in handy.

DEF: Given a real number x : The floor of x is the biggest integer which is smaller or equal to x The ceiling of x is the smallest integer greater or equal to x.

NOTATION: floor(x) = x , ceiling(x) = x Q: Compute 1.7, -1.7, 1.7, -1.7.

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Ceiling and Floor

A: 1.7 = 1, -1.7 = -2, 1.7 = 2, -1.7 = -1

Prove : show that for all positive real numbers x, y:

x.y <= x . y