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Functions. CS/APMA 202 Rosen section 1.8 Aaron Bloomfield. Definition of a function. A function takes an element from a set and maps it to a UNIQUE element in another set. Function terminology. f maps R to Z. f. R. Z. Co-domain. Domain. f(4.3). 4. 4.3. Pre-image of 4.

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Functions

Functions

CS/APMA 202

Rosen section 1.8

Aaron Bloomfield


Definition of a function
Definition of a function

  • A function takes an element from a set and maps it to a UNIQUE element in another set


Function terminology
Function terminology

f maps R to Z

f

R

Z

Co-domain

Domain

f(4.3)

4

4.3

Pre-image of 4

Image of 4.3


More functions

“a”

“bb“

“cccc”

“dd”

“e”

1

2

3

4

5

Alice

Bob

Chris

Dave

Emma

A

B

C

D

F

A string length function

A class grade function

More functions

The image

of A

A pre-image

of 1

Domain

Co-domain


Even more functions

a

e

i

o

u

1

2

3

4

5

“a”

“bb“

“cccc”

“dd”

“e”

1

2

3

4

5

Some function…

Even more functions

Range

Not a valid function!

Also not a valid function!


Function arithmetic
Function arithmetic

  • Let f1(x) = 2x

  • Let f2(x) = x2

  • f1+f2 = (f1+f2)(x) = f1(x)+f2(x) = 2x+x2

  • f1*f2 = (f1*f2)(x) = f1(x)*f2(x) = 2x*x2 = 2x3


One to one functions

a

e

i

o

1

2

3

4

5

a

e

i

o

1

2

3

4

5

A one-to-one function

A function that is

not one-to-one

One-to-one functions

  • A function is one-to-one if each element in the co-domain has a unique pre-image


More on one to one

a

e

i

o

1

2

3

4

5

A one-to-one function

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

e

i

o

u

1

2

3

4

a

e

i

o

1

2

3

4

5

An onto function

A function that is not onto

Onto functions

  • A function is onto if each element in the co-domain is an image of some pre-image


More on onto

a

e

i

o

u

1

2

3

4

An onto function

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

a

b

c

1

2

3

4

a

b

c

d

1

2

3

a

b

c

d

a

b

c

d

1

2

3

4

1

2

3

4

a

b

c

1

2

3

4

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

a

b

c

d

1

2

3

4

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
Identity functions

  • A function such that the image and the pre-image are ALWAYS equal

  • f(x) = 1*x

  • f(x) = x + 0

  • The domain and the co-domain must be the same set


Inverse functions
Inverse functions

Let f(x) = 2*x

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

a

b

c

1

2

3

4

a

b

c

d

1

2

3

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
Compositions of functions

  • Let (f ○ g)(x) = f(g(x))

  • Let f(x) = 2x+3 Let g(x) = 3x+2

  • g(1) = 5, f(5) = 13

  • Thus, (f ○ g)(1) = f(g(1)) = 13


Compositions of functions1
Compositions of functions

f ○ g

A

B

C

g

f

g(a)

f(a)

a

f(g(a))

g(a)

(f ○ g)(a)


Compositions of functions2
Compositions of functions

Let f(x) = 2x+3 Let g(x) = 3x+2

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(3x+2)+3 = 6x+7


Compositions of functions3
Compositions of functions

Does f(g(x)) = g(f(x))?

Let f(x) = 2x+3 Let g(x) = 3x+2

f(g(x)) = 2(3x+2)+3 = 6x+7

g(f(x)) = 3(2x+3)+2 = 6x+11

Function composition is not commutative!

Not equal!


Graphs of functions
Graphs of functions

f(x)=3

x=1

Let f(x)=2x+1

Plot (x, f(x))

This is a plot

of f(x)

f(x)=5

x=2


Useful functions
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) = floor(x+0.5)


Rosen question 8 1 8
Rosen, question 8 (§1.8)

  • Find these values

  • 1.1

  • 1.1

  • -0.1

  • -0.1

  • 2.99

  • -2.99

  • ½+½

  • ½ + ½ + ½

1

2

-1

0

3

-2

½+1 = 3/2 = 1

0 + 1 + ½ = 3/2 = 2


Ceiling and floor properties
Ceiling and floor properties

Let n be an integer

(1a) x = n if and only if n ≤ x < n+1

(1b) x = n if and only if n-1 < x ≤ n

(1c) x = n if and only if x-1 < n ≤ x

(1d) x = n if and only if x ≤ n < x+1

(2) x-1 < x ≤ x ≤ = x < x+1

(3a) -x = - x

(3b) -x = - x

(4a) x+n = x+n

(4b) x+n = x+n


Ceiling property proof
Ceiling property proof

  • Prove rule 4a: x+n = x+n

    • Where n is an integer

    • Will use rule 1a: x = n if and only if n ≤ x < n+1

  • Direct proof!

    • Let m = x

    • Thus, m ≤ x < m+1 (by rule 1a)

    • Add n to both sides: m+n ≤ x+n < m+n+1

    • By rule 4a, m+n = x+n

    • Since m = x, m+n also equals x+n

    • Thus, x+n = m+n = x+n


Factorial
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
Proving function problems

  • Rosen, question 32, §1.8

  • Let f be a function from A to B, and let S and T be subsets of A. Show that


Proving function problems1
Proving function problems

  • Rosen, question 32 (a): f(SUT) = f(S) U f(T)

  • Will show that each side is a subset of the other

  • Two cases!

  • Show that f(SUT)  f(S) U f(T)

    • Let b  f(SUT). Thus, b=f(a) for some aS U T

    • Either aS, in which case bf(S)

    • Or aT, in which case bf(T)

    • Thus, bf(S) U f(T)

  • Show that f(S) U f(T)  f(S U T)

    • Let b  f(S) U f(T)

    • Either b  f(S) or b  f(T) (or both!)

    • Thus, b = f(a) for some a  S or some a  T

    • In either case, b = f(a) for some a  S U T


Proving function problems2
Proving function problems

  • Rosen, question 32 (b): f(S∩T)  f(S) ∩ f(T)

  • Let b  f(S∩T). Then b = f(a) for some a  S∩T

  • This implies that a  S and a  T

  • Thus, b  f(S) and b  f(T)

  • Therefore, b  f(S) ∩ f(T)


Proving function problems3
Proving function problems

  • Rosen, question 62, §1.8

  • 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


Proving function problems4
Proving function problems

  • Rosen, question 62, §1.8

  • Thus, we want to show, for all zZ and xX

  • The second equality is similar


Quick survey
Quick survey

  • I felt I understood the material in this slide set…

  • Very well

  • With some review, I’ll be good

  • Not really

  • Not at all


Quick survey1
Quick survey

  • The pace of the lecture for this slide set was…

  • Fast

  • About right

  • A little slow

  • Too slow


Quick survey2
Quick survey

  • How interesting was the material in this slide set? Be honest!

  • Wow! That was SOOOOOO cool!

  • Somewhat interesting

  • Rather borting

  • Zzzzzzzzzzz


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