1. Overview of Sets an Functions for ICS 6D
Prof. Sandy Irani

2. Sets A set is an unordered collection of items.
For example, S = {a, b, c, d}
Curly braces {} denote that order does not matter:
{a, b, c, d} = {b, a, d, c}
Each item is called an element of the set.
b is an element of S (b ∈ S)
e is not an element of S (e ∉ S)

3. Cardinality of Sets
An infinite set has an infinite number of elements.
Example: the set of all integers.
A finite set has a finite number of elements.
Example: the set of students enrolled in ICS 6D Spr 2016.
If S is a finite set, then the cardinality of S (denoted |S|) is the number of elements in S.
Example: S = {a, b, c, d}. |S| =

4. Famous Sets ℤ = the set of all integers ℝ = the set of real numbers
ℚ = the set of rational numbers
(A number x is rational if x = c/d, where c and d are integers and d ≠ 0.)
ℕ = natural numbers (positive integers)
 the empty set (sometimes denoted as {})

5. Specifying a Set Roster notation: List the elements with curly braces
{1, 3, 5, 9}
List elements with an inferred pattern in ellipses
{1, 3, 5, …., 99}
Set builder notation
{x : x ∈ S and some additional conditions on x}
{x ∈ S : additional conditions on x}
S is a larger set that has already been defined
“:” is read as “such that”

6. Subsets T is a subset of S (T ⊆ S): To show T ⊈ S, Example:
If x ∈ T then x ∈ S
To show T ⊈ S,
Find x ∈ T and x ∉ S.
Example:
S = {a, b, c, d}
T = {a, b, c}
V = {a, e}

7. Set Operations Union: x ∈ A ∪ B ↔ x ∈ A ∨ x ∈ B Intersection:
Complement:
x ∈ A ↔ (x ∈ A)
(all elements and sets contained in a
Universe set, usually denoted by U)

8. Set Operations Example
A = { x ∈ ℤ : x is odd } U = ℤ
B = { x ∈ ℤ : 0 < x 20 }
C = {4, 5, 6, 7}
A ∩ B
C ∩ A
C ⊆ B?
B ⊆ A?
6 ∈ A ∪ C ?
26 ∈ A ∪ C ?

9. Power Set
Let A be a finite set. Power set of A (denoted P(A)) is the set of all subsets of A.
Example: A = {a, b, c}
P(A) = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }
{a, b} ∈ P(A) ?
{ {a} } ⊆ P(A) ?
{a, b} ⊆ P(A) ?
Ø ⊆ P(A) ?
a ∈ P(A) ?
Ø ∈ P(A) ?
{a} ∈ P(A) ?

10. Pairs, Triplets and Tuples
(a, b) is an ordered pair.
Parens (as opposed to {}) indicate that order matters:
(a, b) ≠ (b, a)
{a, b} = {b, a}
(a, b, c) is an ordered triplet
b is the second entry of the triplet (a, b, c)
(a, b, c, d) is an ordered 4-tuple
(a1, a2 , …, an) is an ordered n-tuple.

11. Cartesian Product S x T = { (s, t) : s ∈ S and t ∈ T }
Let S and T be sets
Cartesian product of S and T is
S x T = { (s, t) : s ∈ S and t ∈ T }
Example: S = {a, b, c} T = {1, 2}
S x T = { (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2) }

12. Cartesian Product T = {1, 2}
T x T = T2 = { (1, 1), (1, 2), (2, 1), (2, 2) }
T ⊆ T2 ?
What is ℝ x ℝ? ℤ x ℤ?

13. Cartesian Product A1, …, An sets:
Example: Drink = {OJ, Coffee}
Main = {Waffles, Eggs, Pancakes}
Side = {Hash browns, Toast}
Breakfast Selections = Drink x Main x Side
(OJ, Eggs, Toast) ∈ Drink x Main x Side
A1, …, An sets:
A1x … x An = { (a1, …, an) : ai in Ai for 1 ≤ i ≤ n }

14. Cartesian Product Let S be a set:
Sn = S x S x … x S = { (s1, .., sn) : each si in S, for 1 ≤ i ≤ n }
Example: {0, 1}5
Example: ℝ4

15. N-tuples and Strings
If  is a set of single characters, elements in n can be denoted without the punctuation, in which case they are called strings.
Example:  = {a, b}
(a, b, a, b) ∈  (denoted as an n-tuple)
abab ∈  (denoted as a string)
{0, 1}3 = set of all binary strings with 3 bits:
{0, 1}3 = { 000, 001, 010, 011, 100, 101, 110, 111 }
n-tuple punctuation is important if the underlying set is not a set of single characters!

16. Strings x  = x = x Example: |abba| = 4. Concatenation:
x = abba y = bab
Concatenation of x and y is xy = abbabab
Concatenation of x and a is abbaa
Empty string  has no characters:
x  = x = x
The length of a string x (denoted by |x|) is the number of characters in the string:
Example: |abba| = 4.

17. Infinite sets of strings
The set of all strings of any length over an alphabet :
* = 0 ∪ 1 ∪ 2 ∪ …..
Example: {0, 1}* = {, 0, 1, 00, 01, 10, 11, 000,….}
+ = 1 ∪ 2 ∪ 3 ∪ …..
Example: {0, 1}+ = {0, 1, 00, 01, 10, 11, 000,….}

18. Functions f: A → B A function maps elements of one set onto another:
A is the domain
A = {a, b, c, d}
B is the target
B = {1, 2, 3, 4, 5} 1 a 2 b
A function maps each element
of the domain to a unique
element in the target set. 3 c 4 d 5

19. Functions f: A → B A function maps elements of one set onto another:
A is the domain
A = {a, b, c, d}
B is the target
B = {1, 2, 3, 4, 5} 1 a 2 b
The range is the set of elements
y in the target for which there is
an element x in the domain such
that f(x) = y. 3 c 4 d 5

20. Functions on ℝ specified by an explicit formula
f(x) = x2 - 4x + 3
Examples of non-functions:
f(x) = ±√x
f(x) = 2/x

21. Functions: one-to-one
A function f: D → T is one-to-one if no two elements in the domain map on to the same element in the target:
∀ x ∈ D, x’ ∈ D, (x ≠ x’) → f(x) ≠ f(x’) 1 a 1 a 2 b 2 b 3 c 3 c 4 d 4 d 5 5

22. One-to-one Examples f: ℝ → ℝ f(x) = x2 f: ℤ → ℤ f(x) = 2x + 3
f: {0, 1}3 → {0, 1}3 replace the last bit with 0, f(111) = 110
f: {0, 1}3 → {0, 1}4 add a 0 to the end f(101) = 1010
A = {a, b, c}
f: P(A) → ℤ. For X ⊆ A, f(X) = |X|

23. One-to-one examples f: {0, 1}3 → {0, 1}2 drop the last bit f(101) = 10
000
001 00
010 01
If f: D → T is
one-to-one,
then |D| ≤ |T|
011 10
100 11
101
If f: D → T and
|D| > |T|
The f can not be
one-to-one.
110
111

24. Functions: onto
A function f: D → T is onto if every element in the target is mapped to by some element in the domain
For every y ∈ T, there is an x ∈ D, such that f(x) = y a 1 a 1 b 2 b 2 c 3 c 3 d d

25. Onto Examples f: ℝ → ℝ f(x) = x2 f: ℤ → ℤ f(x) = 2x + 3
f: {0, 1}3 → {0, 1}3 replace the last bit with 0, f(111) = 110
f: {0, 1}3 → {0, 1}2 drop the last bit f(101) = 10
f: {0, 1}3 → {0, 1}3 remove the last bit and concatenate it at the beginning of the string:
f(101) = f(100) = 010

26. Onto Examples If f:D →T is onto, then |T| ≤ |D| If f:D →T and
Example f: {0, 1}2 → {0, 1}3
add a 0 to the end f(10) = 100
000
001 00
010
If f:D →T is
onto,
then |T| ≤ |D| 01
011 10
100 11
101
If f:D →T and
|T| > |D|
The f can not be
onto.
110
111

27. Bijections Definition: A function f:D→T is a bijection if it is
one-to-one and onto a 1 b 2 c 3 d 4
If f:D→T and
f is a bijection, then
|D| = |T|

28. Inverse of a function A function f is a bijection if and only if
f: D → T
The inverse of f (if it exists) is a function f-1: T → D
For every x ∈ D and y ∈ T, f(x) = y ↔ f-1(y) = x a 1 1 a
A function f is a
bijection if and only if
f has an inverse b 2 2 b c 3 3 c d 4 4 d f
f-1

29. Inverse of a function example
A string is a palindrome if it is the same after it is reversed.
Let P6 be the set of all 6-bit strings that are also palindromes.
Bijection between {0, 1}3 and P6
f: {0, 1}3 → P6 f(x) = xxR (xR is the reverse of x)