Set theory

This Power point presentation is prepared to make the concepts of set more easy. Hope you will be helpful for improving your concepts.

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, ………….}

17. 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 

18. Subset OR Proper Subset What is Subset?? https://www.youtube.com/watch?v=_9Wvu-R04go&list=PLmdFyQYShrjfi7EeDyHxr0jhoPXEOlFX0&index=15 Difference b/w Subset & Proper Subset??? https://www.youtube.com/watch?v=xotLg-oLboY

19. 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 }

20. 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 = 2 n In general, n[P(A)] = 2 n where n is the number of elements in set A.