1. CS100: Discrete structures
Computer Science Department
Lecture 1: Set and Sets Operations

2. Lecture Contents Sets Definition. Some Important Sets.
Notation used to describe membership in sets.
How to describe a set?
Sets.
Venn diagrams.
Subset.
Finite and Infinite Sets.
Cardinality.
Sets Operations.
Exercises.
12-Nov-18
Computer Science Department

3. Sets and sets operations
12-Nov-18
Computer Science Department

4. Sets Definition:
Set is the fundamental discrete structure on which all other discrete structures are built.
Sets are used to group objects together. Often, the objects in a set have similar properties.
A set is an unordered collection of objects.
The objects in a set are called the elements or members of the set
12-Nov-18
Computer Science Department

5. Some Important Sets: The set of natural numbers: The set of integers:
Z = {. . . ,−2,−1, 0, 1, 2, . . .}
The set of positive integers:
Z+ = {1, 2, 3, . . .}
The set of fractions:
Q = {0,½, –½, –5, 78/13,…}
Q ={p/q | pЄ Z , qЄZ, and q≠0 }
The set of Real:
R = {–3/2,0,e,π2,sqrt(5),…}
12-Nov-18
Computer Science Department

6. Notation used to describe membership in sets
a set A is a collection of elements.
If x is an element of A, we write xA; If not: xA.
xA Say: “x is a member of A” or “x is in A”.
Note: Lowercase letters are used for elements, capitals for sets.
Two sets are equal if and only if they have the same elements
A= B : x( x A  x B) also
Two sets A and B are equal if A  B and B  A.
So to show equality of sets A and B, show:
A  B
B  A
12-Nov-18
Computer Science Department

7. Notation used to describe membership in sets
The sets {1,3,5} and {3,5,1} are equal , because they have the same elements.
Is {1,3,3,3,5,5,5,5} = {1,3,5} ?!
Yes , because they have the same elements.
12-Nov-18
Computer Science Department

8. How to describe a set?
List all the members of a set, when this is possible. We use a notation where all members of the set are listed between braces. { }
Example :
{dog, cat, horse}
The set O of odd positive integers less than 10 can be expressed by
O={1,3,5,7,9}
12-Nov-18
Computer Science Department

9. How to describe a set?
Sometimes the brace notation is used to describe a set without listing all its members. Some members of the set are listed, and then ellipses (...) are used when the general pattern of the elements is obvious.
Example:
The set A of positive integers less than 100 can be denoted by
A={1, 2, 3, , 99}
12-Nov-18
Computer Science Department

10. How to describe a set?
Another way to describe a set is to use set builder notation. We characterize all those elements in the set by stating the property or properties they must have to be members.
Example:
the set O of all odd positive integers less than 10 can be written as:
O = {x | x is an odd positive integer <10} or, specifying the universe as the set of positive integers, as
O = {x  Z+ | x is odd and x<10}. 
12-Nov-18
Computer Science Department

11. Sets: The Empty Set (Null Set)
We use  to denote the empty set and can also be denoted { }, i.e. the set with no elements.
Example:
the set of all positive integers that are greater than their squares is the null set.
Singleton set
A set with one element is called a singleton set.
12-Nov-18
Computer Science Department

12. Sets: Computer Science
Note that the concept of a data type, or type, in computer science is built upon the concept of a set. In particular, a data type is the name of a set, together with a set of operations that can be performed on objects from that set.
Example:
Boolean is the name of the set {0, 1} together with operators on one or more elements of this set, such as AND, OR, and NOT.
12-Nov-18
Computer Science Department

13. Venn diagrams:
Sets can be represented graphically using Venn diagrams.
In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle.
Inside this rectangle, circles or other geometrical figures are used to represent sets.
Sometimes points are used to represent the particular elements of the set.
12-Nov-18
Computer Science Department

14. Venn diagrams: Example:
A Venn diagram that represents V = {a, e, i, o, u} the set of vowels in the English alphabet
12-Nov-18
Computer Science Department

15. Subset:
The set A is said to be a subset of B if and only if every element of A is also an element of B.
We use the notation A  B to indicate that A is a subset of the set B.
We see that A  B if and only if
the quantification
x (x  A → x  B) is true.
Examples:
The set of all odd positive integers less than 10 is subset of the set of all positive integers .
The set of rational numbers is subset of the set of real numbers .
12-Nov-18
Computer Science Department

16. x ((x  A)  (x  B))  x ((x  B)  (x  A))
Subsets:
For every set S,
  S
S  S
Proper subset:
When a set A is a subset of a set B but A ≠ B,
A  B, and A  B
We write A  B and say that A is a proper subset of B
For A  B to be true, it must be the case that
x ((x  A)  (x  B))  x ((x  B)  (x  A))
12-Nov-18
Computer Science Department

17. Subsets: Quick Examples: {1,2,3}  {1,2,3,4,5} {1,2,3}  {1,2,3,4,5}
Is   {1,2,3}?
Is   {1,2,3}?
Is   {,1,2,3}?
Is   {,1,2,3}?
Yes!
No!
Yes!
Yes!
12-Nov-18
Computer Science Department

18. Subsets: Quiz Time: Is {x}  {x,{x}}? Is {x}  {x,{x}}? Is {x}  {x}?
Yes!
Yes!
Yes!
No!
12-Nov-18
Computer Science Department

19. Finite and Infinite Sets:
Let S be a set. If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is a finite set and that n is the cardinality of S.
The cardinality of S is denoted by |S|.
| A  B | = | A| + | B| - | A  B|  
Infinite set
A set is said to be infinite if it is not finite. For example, the set of positive integers is infinite.
N.B. We only count unrepeated elements
12-Nov-18
Computer Science Department

20. Cardinality: Find S = {1,2,3}, S = {3,3,3,3,3}, S = ,
|S| is infinite
12-Nov-18
Computer Science Department

21. Sets: Ways to Define Sets: Explicitly: {John, Paul, George, Ringo}
Implicitly: {1,2,3,…}, or {2,3,5,7,11,13,17,…}
Set builder: { x : x is prime }, { x | x is odd }.
In general { x : P(x) is true }, where P(x) is some description of the set.
12-Nov-18
Computer Science Department

22. The power of a set:
Many problems involve testing all combinations of elements of a set to see if they satisfy some property.
To consider all such combinations of elements of a set S, we build a new set that has as its members all the subsets of S.
Given a set S, the power set of S is the set of all subsets of the set S.
The power set of S is denoted by P(S).
if a set has n elements , then the power has 2n elements.
12-Nov-18
Computer Science Department

23. The power of a set: Example:
What is the power set of the set {0, 1, 2}?
P({0,1,2}) is the set of all subsets of {0, 1, 2}
P({0,1,2})= { , {0},{1},{2},{0,1},{0,2},{1,2},{0,1,2}}
What is the power set of the empty set? What is the power set of the set {} ?
P()= {}
P({})= {,{}}
N.B. the power set of any subset has at least two elements
The null set and the set itself
12-Nov-18
Computer Science Department

24. P(S) = {, {}, {{}}, {,{}}}.
The Power Set:
Quick Quiz:
Find the power set of the following:
S = {a},
S = {a,b},
S = ,
S = {,{}},
P(S)= {, {a}}.
P(S) = {, {a}, {b}, {a,b}}.
P(S) = {}.
P(S) = {, {}, {{}}, {,{}}}.
12-Nov-18
Computer Science Department

25. Cartesian Products:
The order of elements in a collection is often important.
Because sets are unordered, a different structure is needed to represent ordered collections.
This is provided by ordered n-tuples.
The ordered n-tuple (a1, a2, , an) is the ordered collection that has a1 as its first element, a2 as its second element, , and an as its nth element.
12-Nov-18
Computer Science Department

26. Cartesian Products:
Let A and B be sets. The Cartesian product of A and B, denoted by A×B, is the set of all ordered pairs (a, b), where aA and bB.
A×B = {(a, b) | a  A  b  B}.
A1×A2×…×An={(a1, a2,…, an) | aiAi for i=1,2,…,n}.
A×B not equal to B×A
Example :
A={1,2} , B={3,4}
A×B={(1,3),(1,4),(2,3),(2,4)}
B×A={(3,1),(3,2),(4,1),(4,2)}
12-Nov-18
Computer Science Department

27. Cartesian Products:
Example:
What is the Cartesian product A × B × C, where
A = {0, 1}, B = {1, 2}, and C = {0, 1, 2}?
AxBxC = {(0,1,0), (0,1,1), (0,1,2), (0,2,0), (0,2,1), (0,2,2), (1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}
12-Nov-18
Computer Science Department

28. Sets and sets operations
12-Nov-18
Computer Science Department

29. UNION: A  B = { x : x  A v x  B} The union of two sets A and B is:
If A = {1, 2, 3}, and B = {2, 4}, then
A  B = {1,2,3,4} A B
12-Nov-18
Computer Science Department

30. Intersection: A  B = { x : x  A  x  B}
The intersection of two sets A and B is:
A  B = { x : x  A  x  B}
If A = {Charlie, Lucy, Linus}, and B = {Lucy, Desi}, then
A  B = {Lucy} A B
12-Nov-18
Computer Science Department

31. Intersection: If A = {x : x is a US president}, and
B = {x : x is deceased}, then
A  B = {x : x is a deceased US president} A B
12-Nov-18
Computer Science Department

32. Sets whose intersection
Disjoint:
If A = {x : x is a US president}, and B = {x : x is in this room}, then A  B = {x : x is a US president in this room} =
Sets whose intersection
is empty are
called disjoint sets A B
12-Nov-18
Computer Science Department

33. Complement: The complement of a set A is: A = A’ = { x : x  A}
If A = {x : x is bored}, then
A = {x : x is not bored} = 
A  B = B  A
= U
and
U =  A U
12-Nov-18
Computer Science Department

34. Difference: The set difference, A - B, is:
A - B = { x : x  A  x  B }
A - B = A  B A U B
12-Nov-18
Computer Science Department

35. Symmetric Difference:
The symmetric difference, A  B, is:
A  B = { x : (x  A  x  B) v (x  B  x  A)}
= (A - B)  (B - A)
Like
“exclusive or” A U B
12-Nov-18
Computer Science Department

36. Symmetric Difference:
Example:
Let A = {1,2,3,4,5,6,7} B = {3,4,p,q,r,s}
Then we have
A  B =
{1,2,3,4,5,6,7,p,q,r,s}
A  B =
{3,4}
We get A  B = {1,2,5,6,7,p,q,r,s}
12-Nov-18
Computer Science Department

37. TABLE 1: Set Identities Name Identity Identity laws A U  = A
Domination laws
A U U = U
A   = 
Idempotent laws
A  A = A
A  A = A
Complementation laws
(A) = A
Commutative laws
A  B = B  A
A  B = B  A
Associative laws
A  (B  C) = (A  B)  C
A  (B  C) = (A  B)  C
Distributive laws
A  (B U C) = (A  B)  (A  C)
A  (B  C) = (A U B)  (A U C)
De Morgan’s laws
A U B = A  B
A  B = A U B
Absorption laws
A  (A  B) = A
A  (A  B) = A
Complement laws
A  A = U
A  A = 
12-Nov-18
Computer Science Department

38. Let’s proof one of the Identities Using a Membership Table A  (B  C) = (A  B)  (A  C)
TABLE 2: A Membership Table for the Distributive Property
(A  B)  (A  C)
A  C
A  B
A  (B  C)
B  C C B A 1
12-Nov-18
Computer Science Department

39. Exercise 1: List the members of these sets:
a) {x | x is a real number such that x² = 1}
{-1, 1}
b) {x | x is a positive integer less than 12}
{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
c) {x | x is the square of an integer and x < 100}
{0, 1, 4, 9, 16, 25, 36, 49, 64, 81}
d) {x | x is an integer such that x² = 2} 
12-Nov-18
Computer Science Department

40. Exercise 2: Determine whether each of these pairs of sets are equal:
Yes
b) {{1}}, {1, {2}} No
c) , {}
12-Nov-18
Computer Science Department

41. Exercise 3: Determine whether these statements are true or false.
b) ∅∈{0}
c) {0} ⊂∅
d) ∅⊂ {0}
e) {0} ∈ {0}
f) {0} ⊂ {0}
g) {∅}⊆{∅}
False
False
False
True
False
False
True
12-Nov-18
Computer Science Department

42. Exercise 4:
Use a Venn diagram to illustrate the relationships 𝐀⊂𝑩 𝒂𝒏𝒅 𝑩⊂𝑪 . A B C U
12-Nov-18
Computer Science Department

43. Exercise 5: What is the cardinality of each of these sets? a) {a} 1
b) {{a}}
c) {∅, {∅}} 2
d) {a, {a}, {a, {a}}} 3
12-Nov-18
Computer Science Department

44. Exercise 6: Let A = {1, 2, 3, 4, 5} and B = {0, 3, 6}. Find : a) A ∪ B
{0, 1, 2, 3, 4, 5, 6}
b) A ∩ B
{3}
c) A – B
{1, 2, 4, 5}
d) B – A
{0, 6}
12-Nov-18
Computer Science Department

45. Exercise 7:
For U = {1, 2,3, 4,5,6,7,8,9,10} let A = {1, 2,3,4,5} , B = {1,2, 4,8}, C = {1, 2,3,5,7}, and D = {2, 4,6,8} . Determine each of the following:
a) (A∪B)∩C =
b) A∪(B∩C)=
c) C ∪ D =
d) (A∪B)−C =
e) A∪(B−C)=
f) (B −C)−D =
g) B−(C−D)=
h) (A∪B)−(C ∩D)=
i) A ⊕ B =
{1,2,3,5}
{1,2,3,4,5}
{1, 2,3, 4, 5, 6, 7, 8}
{4,8}
{1,2,3,4,5,8} {}
{2,4,8}
{1,3,4,5,8}
{3,5,8}
12-Nov-18
Computer Science Department

46. Exercise 8:
Draw the VENN DIAGRAM of these sets and find (A∪B)−C and B′ A B C B
12-Nov-18
Computer Science Department

47. Exercise 9:
Given the Universal set U={positive integers not larger than 12}, and the sets : A={positive integers not more than 6}
B={3,4,6,7} , C={5,6,7,8,9,10} , Find :
i) A U B =
ii) | A−B |=
iii) P(A‐B)=Power set of (A‐B)=
{1,2,3,4,5,6,7} 3
{ф,{1},{2},{5},{1,2},{1,5},{2,5},{1,2,5}}
12-Nov-18
Computer Science Department

48. Refer to Chapter 2 for further reading
12-Nov-18
Computer Science Department