1. Functions

2. Functions
A function f from a set A to a set B is an assignment 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
(note: Here, ““ has nothing to do with if… then)

3. Functions
If f:A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 the pre-image of b.
The range of f:AB is the set of all images of elements of A.
We say that f:AB maps A to B.

4. Function terminology f maps R to Z R f Z Domain Co-domain f(4.3) 4.3 4
Pre-image of 4
Image of 4.3

5. A string length function
More functions
The image
of A
A pre-image
of 1
Domain
Co-domain A B C D F
Alice
Bob
Chris
Dave
Emma
A class grade function 1 2 3 4 5
“a”
“bb“
“cccc”
“dd”
“e”
A string length function

6. Functions Let us take a look at the function f:PC with
P = {Linda, Max, Kathy, Peter}
C = {Boston, New York, Hong Kong, Moscow}
f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = New York
Here, the range of f is C.

7. Functions Other ways to represent f: x f(x) Linda Moscow Max Boston
Kathy
Hong Kong
Peter
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow

8. Operations on Functions
Let f1 and f2 be functions from A to R.
Then the sum and the product of f1 and f2 are also functions from A to R defined by:
(f1 + f2)(x) = f1(x) + f2(x)
(f1f2)(x) = f1(x) f2(x)
Example:
f1(x) = 3x, f2(x) = x + 5
(f1 + f2)(x) = f1(x) + f2(x) = 3x + x + 5 = 4x + 5
(f1f2)(x) = f1(x) f2(x) = 3x (x + 5) = 3x2 + 15x

9. Functions
We already know that the range of a function f:AB is the set of all images of elements aA.
If we only regard a subset SA, the set of all images of elements sS is called the image of S.
We denote the image of S by f(S):
f(S) = {f(s) | sS}

10. Functions Let us look at the following well-known function:
f(Linda) = Moscow
f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Boston
What is the image of S = {Linda, Max} ?
f(S) = {Moscow, Boston}
What is the image of S = {Max, Peter} ?
f(S) = {Boston}

11. One-to-one functions
A function is one-to-one if each element in the co-domain has a unique pre-image
Meaning no 2 values map to the same result 1 2 3 4 5 a e i o
A one-to-one function 1 2 3 4 5 a e i o
A function that is
not one-to-one

12. 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 1 2 3 4 5 a e i o
A one-to-one function

13. 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
Meaning all elements in the right are mapped to 1 2 3 4 a e i o u
An onto function 1 2 3 4 5 a e i o
A function that is not onto

14. 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 1 2 3 4 a e i o u
An onto function

15. Onto vs. one-to-one
Are the following functions onto, one-to-one, both, or neither? 1 2 3 4 a b c 1 2 3 4 a b c d 1 2 3 4 a b c
1-to-1, not onto
Both 1-to-1 and onto
Not a valid function 1 2 3 a b c d 1 2 3 4 a b c d
Onto, not 1-to-1
Neither 1-to-1 nor onto

16. Bijections Consider a function that is both one-to-one and onto:1 2 3 4 a b c d
Consider a function that is both one-to-one and onto:
Such a function is a one-to-one correspondence, or a bijection

17. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Is f injective?
No.
Is f surjective?
Is f bijective?

18. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Is f injective?
No.
Is f surjective?
Yes.
Is f bijective?
Paul

19. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Lübeck
Is f injective?
Yes.
Is f surjective?
No.
Is f bijective?

20. Properties of Functions
Linda
Max
Kathy
Peter
Boston
New York
Hong Kong
Moscow
Lübeck
Is f injective?
No! f is not even a function!

21. Properties of Functions
Linda
Boston
Is f injective?
Yes.
Is f surjective?
Is f bijective?
Max
New York
Kathy
Hong Kong
Peter
Moscow
Helena
Lübeck

22. 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

23. Inverse functions Let f(x) = 2*x R f R f-1 f(4.3) 4.3 8.6 f-1(8.6)
Then f-1(x) = x/2

24. Inversion
An interesting property of bijections is that they have an inverse function.
The inverse function of the bijection f:AB is the function f-1:BA with
f-1(b) = a whenever f(a) = b.

25. Inversion Linda Boston f Max New York f-1 Kathy Hong Kong Peter Moscow
Helena
Lübeck

26. Inversion Example: f(Linda) = Moscow f(Max) = Boston
f(Kathy) = Hong Kong
f(Peter) = Lübeck
f(Helena) = New York
Clearly, f is bijective.
The inverse function f-1 is given by:
f-1(Moscow) = Linda
f-1(Boston) = Max
f-1(Hong Kong) = Kathy
f-1(Lübeck) = Peter
f-1(New York) = Helena
Inversion is only possible for bijections (= invertible functions)

27. Inversion
Linda
Boston f
Max
New York
f-1
f-1:CP is no function, because it is not defined for all elements of C and assigns two images to the pre-image New York.
Kathy
Hong Kong
Peter
Moscow
Helena
Lübeck

28. 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

29. Compositions of functions
f ○ g A B C g f
g(a)
f(a) a
f(g(a))
g(a)
(f ○ g)(a)

30. 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

31. 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!

32. Graphs of functions Let f(x)=2x+1 Plot (x, f(x)) This is a plot
of f(x)
f(x)=5
x=2

33. 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)

34. Floor and Ceiling Functions
The floor and ceiling functions map the real numbers onto the integers (RZ).
The floor function assigns to rR the largest zZ with zr, denoted by r.
Examples: 2.3 = 2, 2 = 2, 0.5 = 0, -3.5 = -4
The ceiling function assigns to rR the smallest zZ with zr, denoted by r.
Examples: 2.3 = 3, 2 = 2, 0.5 = 1, -3.5 = -3

35. Sample floor/ceiling questions
Find these values
1.1
1.1
-0.1
-0.1
2.99
-2.99
½+½
½ + ½ + ½ 1 2 -1 3 -2
½+1 = 3/2 = 1
 ½ = 3/2 = 2

36. 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

37. 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

38. 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

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

40. Proving function problems
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

41. Proving function problems
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)

42. 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

43. Proving function problems
Thus, we want to show, for all zZ and xX
The second equality is similar

44. Graphs
The graph of a function f:AB is the set of ordered pairs {(a, b) | aA and f(a) = b}.
The graph is a subset of AB that can be used to visualize f in a two-dimensional coordinate system.