SET THEORY Chumki Sarkar Definition A collection of

SET THEORY Chumki Sarkar

Definition: �A collection of objects is defined to be a set when � (i) the collection is well-defined; � (ii) objects belonging to the collection are different; � (iii) objects of the collection are independent of the order of their arrangement Chumki S arkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

METHODS OF SET PRESENTATION Presentation of Set ROSTER or TABULAR METHOD PROPERTY or SET-BUILDER METHOD Chumki Sark ar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

ROSTER METHOD � The set is described by just listing its elements inside brackets. � Ex: the set of all vowels of the English alphabet A={a, e, i, o, u} � Demerits: This method fails if all the elements of the set cannot be displayed. In that case the property method is used. Chumki S arkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

PROPERTY OR SET-BUILDER METHOD � The set is described as A={x/P(x)} where x is any element possessing the property P(x). � Ex: The set of all prime numbers is described as A={x/x is a prime number} Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

SINGLETON SET �A set consisting of only one element is called singleton set. � Ex: A={4} Chumki Sa rkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

NULL SET �A set which contains no elements at all is called null set or empty set or void set. It is denoted by Φ( Phi). � Ex: Φ={x: x is an integer and 2<x<3} Chumk i Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

FINITE AND INFINITE SETS �A set is said to be finite if the number of elements contained in the set can be counted and the counting process has an end. Ex: A={10, 11, 12, 13, 14, 15} i. e. n(A)=6 �A set is said to be infinite if the number of elements contained in the set can not be determined by counting. Ex: A={1, 1/2 , 1/3 , 1/4 , . . . } Chumki S arkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

EQUAL SET � Two sets A & B are said to be equal if all the elements of A are the elements of B as well as all the elements of B are the elements of A and written as A=B Example: A={1, 2, 3, 4} B={2, 4, 1, 3} Chumki Sar kar then A=B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

UNIVERSAL SET �A set U is called Universal set if all the sets under consideration are the subsets of U. Example: U={1, 2, 3, 4, 5, 6, 7, 8, 9} A={2, 4, 6, 8} 5 1 2 3 Chumki Sar kar 8 A 4 : Universal Set : Subset U 7 6 Presented by: Chumki Sarkar, Maheshtala College 9 30 -09 -2020

SUB SET A B Examples: �A “A is a subset of B” if and only if every element of A is also an element of B. = {3, 9}, B = {5, 9, 1, 3}, A B U A Chumki Sark ar B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

DISJOINT SET Two sets A & B are said to be disjoint if they have no common elements. Example: A={1, 2, 3} & B={4, 5, 6} A Ch umki Sar k ar U B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

UNION OF TWO SETS � The Union of two given sets A & B written as A ⋃ B is defined to be the set of all elements which belong either to A or to B or to both. Example: A={1, 2, 3, 4} & B={3, 4, 5, 6} then, A ⋃ B = {1, 2, 3, 4, 5, 6} U A Chumki Sarkar B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

INTERSECTION OF TWO SETS � The intersection of two sets A & B written as A ⋂ B is the set of all elements which belong to both A & B Example: A={1, 2, 3, 4} & B={3, 4, 5, 6} then, A ⋂ B = {3, 4, } U A Chumki Sarkar B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

DIFFERENCE OF TWO SETS � The difference between two sets A & B written as A-B is the set of all elements which belong to A but which do not belong to B Example: A={1, 2, 3, 4} & B={3, 4, 5, 6} then, A - B = {1, 2} U A Chumki Sarkar B Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

COMPLEMENT OF A SET � The compliment of a set A written as is the set of all elements of the universal set U which do not belong to A Example: A={1, 2, 3, 4} & U= {1, 2, 3, 4, 5, 6, 7, 8} then, = {5, 6, 7, 8} U A Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

LAWS OF SET OPERATIONS Commutative Laws Associative Laws Distributive Laws De Morgan’s Laws Chumki Sarkar for any two sets A & B for any 3 sets A, B & C i) A⋃B = B⋃A, ii) A⋂B = B⋂A i) A⋃(B⋃C)=(A⋃B)⋃C, ii) A⋂(B ⋂C)=(A⋂B)⋂C for any 3 sets A, B & C i) A⋃(B⋂C)=(A⋃B)⋂(A⋃C), ii) A⋂(B⋃C)=(A⋂B)⋃(A⋂C) for any two sets A & B i) (A⋃B)C =AC⋂BC ii) (A⋂B)C=AC⋃BC Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Commutative Laws = = Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Associative Laws (i) Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Associative Laws From equation (i) & (ii) Chumki Sarkar (ii) Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Associative Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Associative Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

De Morgan’s Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

De Morgan’s Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020

THANK YOU Chumki Sar kar Presented by: Chumki Sarkar, Maheshtala College 30 -09 -2020