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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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For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

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