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Discrete MathematicsFunctions

Definition of a function

- A function takes an element from a set and maps

it to a UNIQUE element in another set

f maps R to Z

R

Z

Co-domain

Domain

f

f(4.3)

4

4.3

Pre-image of 4

Image of 4.3

More functions

The image of a

A pre-image of 1

Domain

Co-domain

Even more functions

Range

Not a valid function! Also not a valid function!

Function arithmetic

- Let f1(x) 2x
- Let f2(x) x2
- f1f2 (f1f2)(x) f1(x)f2(x) 2xx2
- f1f2 (f1f2)(x) f1(x)f2(x) 2xx2 2x3

One-to-one functions

- A function is one-to-one if each element in the

co-domain has a unique pre-image - Formal definition A function f is one-to-one if

f(x) f(y) implies x y.

More on one-to-one

- Injective is synonymous with one-to-one
- A function is injective
- A function is an injection if it is one-to-one
- Note that there can be un-used elements in the

co-domain

Onto functions

- A function is onto if each element in the

co-domain is an image of some pre-image - Formal definition A function f is onto if for

all y ? C, there exists x ? D such that f(x)y.

More on onto

- Surjective is synonymous with onto
- A function is surjective
- A function is an surjection if it is onto
- Note that there can be multiply used elements

in the co-domain

Onto vs. one-to-one

- Are the following functions onto, one-to-one,

both, or neither?

1-to-1, not onto

Both 1-to-1 and onto

Not a valid function

Onto, not 1-to-1

Neither 1-to-1 nor onto

Bijections

- Consider a function that isboth one-to-one and

onto - Such a function is a one-to-one correspondence,

or a bijection

Identity functions

- A function such that the image and the pre-image

are ALWAYS equal - f(x) 1x
- f(x) x 0
- The domain and the co-domain must be the same set

Inverse functions

Let f(x) 2x

f

R

R

f-1

f(4.3)

8.6

4.3

f-1(8.6)

Then f-1(x) x/2

More on inverse functions

- Can we define the inverse of the following

functions? - An inverse function can ONLY be done defined on a

bijection

What is f-1(2)? Not onto!

What is f-1(2)? Not 1-to-1!

Compositions of functions

(f ? g)(x) f(g(x))

f ? g

A

B

C

g

f

g(a)

f(b)

a

f(g(a))

b g(a)

(f ? g)(a)

Compositions of functions

Let f(x) 2x3 Let g(x) 3x2

f ? g

R

R

R

g

f

g(1)

f(5)

f(g(1))13

1

g(1)5

(f ? g)(1)

f(g(x)) 2(3x2)3 6x7

Compositions of functions

- Does f(g(x)) g(f(x))?
- Let f(x) 2x3 Let g(x) 3x2
- f(g(x)) 2(3x2)3 6x7
- g(f(x)) 3(2x3)2 6x11
- Function composition is not commutative!

Not equal!

Useful functions

- Floor ?x? means take the greatest integer less

than or equal to the number - Ceiling ?x? means take the lowest integer

greater than or equal to the number - round(x) ? x0.5 ?

Ceiling and floor properties

- Let n be an integer
- (1a) ?x? n if and only if n x lt n1
- (1b) ?x? n if and only if n-1 lt x n
- (1c) ?x? n if and only if x-1 lt n x
- (1d) ?x? n if and only if x n lt x1
- (2) x-1 lt ?x? x ?x? lt x1
- (3a) ?-x? - ?x?
- (3b) ?-x? - ?x?
- (4a) ?xn? ?x?n
- (4b) ?xn? ?x?n

Ceiling property proof

- Prove rule 4a ?xn? ?x?n
- Where n is an integer
- Will use rule 1a ?x? n if and only if n x

lt n1 - Direct proof!
- Let m ?x?
- Thus, m x lt m1 (by rule 1a)
- Add n to both sides mn xn lt mn1
- By rule 4a, mn ?xn?
- Since m ?x?, mn also equals ?x?n
- Thus, ?x?n mn ?xn?

Factorial

- Factorial is denoted by n!
- n! n (n-1) (n-2) 2 1
- Thus, 6! 6 5 4 3 2 1 720
- Note that 0! is defined to equal 1

Proving Function problems

- Let f be an invertible function from Y to Z
- Let g be an invertible function from X to Y
- Show that the inverse of f?g is
- (f?g)-1 g-1 ? f-1
- (Pf) Thus, we want to show, for all z?Z and x?X

((f ? g) ? (g-1 ? f-1)) (x) x and ((f-1 ? g-1)

? (g ? f)) (z) z - ((f ? g) ? (g-1 ? f-1)) (x) (f ? g) ?(g-1 ?

f-1)) (x)) - (f ? g) ?g-1 ?f-1(x)))
- (f ?g ?g-1 ?f-1(x)))))
- (f ?f-1(x))
- x
- The second equality is similar

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