The document provides an introduction to the concept of sets in mathematics, covering definitions, types, symbols, and notations such as descriptive, tabular, and set builder notation. It includes examples of various sets, including natural numbers, integers, and subsets, as well as explanations of types of sets like empty, finite, infinite, equal, and equivalent sets. Additionally, it discusses the power set and the calculation of subsets.

1. SET
By: Syed Hasnain Javed Zaidi

2. SET
• Introduction to Set
• Types & Symbols of Number Sets
• Forms of Set (Descriptive, Tabular & Set Builder Notation)
• Types of Set

3. Types of Numbers

4. Symbol of Number Sets
• Natural Numbers N
• Whole Numbers W
• Set of Integers Z
• Set of Negative Integers Z-
• Set of Prime Numbers P
• Set of Even Numbers E
• Set of Odd Numbers O
• Set of Rational Numbers Q
• Set of Irrational Numbers Q’
• Set of Real Numbers R

5. • Descriptive Form
• Tabular Form
• Set Builder Notation
Forms of Set

6. • Example:
{Name of Days in a week } Descriptive Form
(Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,
Sunday}
Tabular Form
{x | x is Names of days in a week} Set Builder Notation

7. • Example:
{Names of Provinces of Pakistan }
Tabular Form
{Sindh, Punjab, KPK, Balochistan}
Set Builder Notation
{x | x is a Province of Pakistan}

8. • Example:
{Set of Natural numbers less than 10 }
Tabular Form
{1, 2, 3, 4, 5, 6, 7, 8, 9}
Set Builder Notation
{x | x€N ˄ x ˂ 10}

9. • Example:
{Set of Whole numbers less than equal to 7 }
Tabular Form
{0, 1, 2, 3, 4,5, 6, 7}
Set Builder Notation
{x | x€W ˄ x ≤ 7}

10. • Example:
{Set of Integers between -100 and 100}
Tabular Form
{-99,-98,-97, ………99}
Set Builder Notation
{x | x ε Z ˄ -100 ˂ x ˂ 100}

11. Write the following in Set Builder form
i) {1, 2, 3, 4,………..100}
{x | x ε N ˄ x ≤ 100}
ii) {0, ±1, ± 2, ± 3, ± 4, ……….. ± 1000}
{x | x ε Z ˄ x ≤ 1000}
iii) { -100, -101, -102, ……….., -500}
{x | x ε Z- ˄ -100 ≤ x ≤ -500}
iv) {January, June July}
{x | x is month start from letter j}

12. v) {0, 1, 2, 3, 4,………..100}
{x | x ε W˄ x ≤ 100}
vi) { -1, - 2, - 3, - 4, ……….. - 500}
{x | x ε Z- ˄ x ≤ -500}
vii) { 100, 101, 102, ……….., 400}
{x | x ε N ˄ 100 ≤ x ≤ 400}
viii) {Peshawar, Lahore, Karachi, Quetta}
{x | x is big cities of Pakistan}

13. Write in Tabular Form
i) {x | x ε N ˄ x ≤10}
{1, 2 , 3, 4, 5, 6, 7, 8, 9, 10}
ii) {x | x ε N ˄ 4 ˂ x ˂12}
{5, 6, 7, 8, 9, 10, 11}
iii) {x | x ε W ˄ x ≤ 4}
{0, 1, 2, 3, 4}
iv) {x | x ε O ˄ 3 ˂ x ˂12}
{5, 7, 9, 11}
v) {x | x ε Z ˄ x + 1 = 0}
{-1}

14. TYPES OF SET
• Empty or Null Set
• Finite Set
• Infinite Set
• Equal Set
• Equivalent Set
• Subset
• Proper Subset
• Improper Subset
• Power Set

15. • Empty Set or Null Set:
Example:
(a) The set of whole numbers less than 0.
(b) A bald student in class 9A
(c) Let A = {x : x ∈ N 2 < x < 3}
(d) Let B = {x : x ∈ Z -1 < x < 0}
Empty or Null Set is denoted by { } or Ø
{Ø} or { 0 } is wrong

16. • Finite Set
• A set which contains a definite (countable) number of elements is called a
finite set. Empty set is also called a finite set.
For example:
• The set of all colors in the rainbow.
• N = {x : x ∈ N, x < 7}
• P = {2, 3, 5, 7, 11, 13, 17, ...... 97}
• Infinite Set
The set whose elements can not be listed (uncountable).
For Example:
• Stars in the sky
• A = {x : x ∈ N, x > 1}
• {0, -1, -2, -3, -4, ………….}

18. • Equivalent Sets:
• Two sets A and B are said to be equivalent if their cardinal number is same, i.e.,
n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’.
For example:
• A = {1, 2, 3} Here n(A) = 3
• B = {p, q, r} Here n(B) = 3
Therefore, A ↔ B
• Equal sets:
• Two sets A and B are said to be equal if they contain the same elements. Every
element of A is an element of B and every element of B is an element of A.
For example:
• A = {p, q, r, s}
B = {p, s, r, q}
Therefore, A = B

19. Subset OR Proper Subset
What is Subset??
https://www.youtube.com/watch?v=_9Wvu-
R04go&list=PLmdFyQYShrjfi7EeDyHxr0jhoPXE
OlFX0&index=15
Difference b/w Subset & Proper Subset???
https://www.youtube.com/watch?v=xotLg-
oLboY

20. Subset & Proper Subsets
Q1. If A = {a, c} Final all possible subsets
{ }, {a}, {c}, {a, c}
Q2. If B = {x, y} Find Proper Subsets
{ }, {x}, {y}

21. • Power Set:
• The collection of all subsets of set A is called the power set of A. It is
denoted by P(A). In P(A), every element is a set.
For example;
• If A = {p, q} then all the subsets of A will be
P(A) = ∅, {p}, {q}, {p, q}
Number of elements of P(A) = n[P(A)] = 4 = 2n
In general, n[P(A)] = 2n where n is the number of elements in set A.