1. Analytical Methods in CS (CIS 505)
Sanchita Mal-Sarkar

2. Sets A set is a collection of objects (duplicates are not allowed).
Objects in the set are called elements or members of the set.
Examples:
Set of students in Discrete Mathematics.
Collection of all positive integers less than 4
A = {1, 2, 3} 1 2 3 A

3. Sets
More Examples:
Collection of all negative integers greater than -6
B = {-5, -4, -3, -2, -1} -2 -1 -3
- 4 A -5

4. Another way to represent sets (Set builder notation)
Sometimes it is impossible/inconvenient to describe a set by listing all its elements.
In general, a set can be defined as:
{x | P(x)}
where P(x) represents a property that the elements of the set have in common. P(x) is a propositional function.
Examples:
Collection of all positive integers
A = {x| x is a positive integer}
B = {x | x is odd OR x is less than 10 }

5. Sets Two sets are equal if they have the same elements. Example:
If A = {1, 2, 3, 4}
B = {x| x is a positive integer and x2 < 20}
then A = B

6. Repeated elements can be ignored
Example:
The set consists of the letter in the word “Book”.
It can be denoted in three ways:
{b, o, o, k}
{b, o, k}
{x| x is a letter in the word “book’}

7. Order is not important Example: Set of all integers less than 4
It can be represented as:
A = {1, 2, 3}
A = {2, 3, 1}
A = {3, 2, 1}

8. x is an element of the set A
x Є A => x is an element of A
Example:
A = {1, 2, 3}
1 Є A
2 Є A
3 Є A
4 Є A
{ } Є A
Ø Є A
A Є A

9. Universal set Set of all possible elements under consideration.
Denoted by U (or sometimes W).
In Venn diagram, the universal set U will be denoted by a rectangle and the sets within U will be denoted by circles. U A

10. Empty/Null Set A set that does not have any element in it.
Denoted by { } or by the symbol Ø.
Examples:
{x| x is a real number and x2 = -1} = Ø
{x| x is a real number and x2 = -9} = Ø
{x| x is a real number and x = x +1} = Ø

11. Subsets A is a subset of B if every element of A is an element of B
{1, 2, 3} is a subset of {1, 2, 3, 4, 5} (in fact a proper subset)
A is a proper subset of B if it is a subset of B but not equal to B.
Sets can have other sets as members –
{ f, {a}, {b}, {a, b}}

12. Cardinality of a set for some finite integer n where n Є N.
A set A is called finite if it has n distinct elements
for some finite integer n where n Є N.
Example of finite set: {1, 2, 3, 4}
A set that is not finite is called infinite.
Example of infinite set: {x| x is a positive integer}
Cardinality of A: Number of distinct elements (n) of a set.
Cardinality of A is denoted by |A|.

13. Power set Power set => Set of all subsets of a set.
If A is a set, then the set of all subsets of A is called the power set of A.
Denoted by P(A)
Note that the empty set is always considered to be a subset of every set. 
Examples:
A = {1,2,3}
P(A) consists of the following subsets of A:
{ }, {1},{2}, {3}, {1,2}, {1,3}, {2,3}, and {1,2,3}
What is the power set of {a, b, c}?

14. Venn Diagram
Venn diagrams are often used to indicate relationships between sets.
If A is a subset of B, we can write A כֿ B B A

15. Venn Diagram
If A is not a subset of B, we can write A כֿ B A B

16. Venn Diagram Every set is a subset of itself. A A כֿ A Ø כֿ
For any set A, Ø A כֿ
Since there are no elements of Ø that are
not in A

17. Venn Diagram
A = B , if and only if A B כֿ
and B A כֿ

18. Venn Diagram {A} כֿ B {{A}} כֿ B However, B A כֿ
Let A be a set and let B ={A,{A}}.
Then, A Є B and {A} Є B.
Then
{A} כֿ B
and
{{A}} כֿ B
However, B A כֿ

19. Venn Diagram
Draw a Venn diagram that represents the following relationships.
x Є A, x Є B, x Є C, y Є B, y Є C, and y Є A A x B
One solution y C

20. Venn Diagram A {2, 3} כֿ {1, 4} A כֿ {1, 2, 3} כֿ A
A = {1, {2, 3}, 4}. Identify whether true or false.
3 Є A
{2, 3} Є A.
{4} Є A A
{2, 3} כֿ
{1, 4} A כֿ
{1, 2, 3} כֿ A