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 Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
METHODS OF SET PRESENTATION Presentation of Set ROSTER or TABULAR METHOD PROPERTY or SET-BUILDER METHOD Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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 Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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 22-02-2019
SINGLETON SET A set consisting of only one element is called singleton set. Ex:A={4} Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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} Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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={10,11,12,12,13,14,15} 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 Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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} then A=B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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} :Universal Set A={2,4,6,8} :Subset 5 U 1 7 4 2 A 3 8 6 9 Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
SUB SET U A B A = {3, 9}, B = {5, 9, 1, 3}, A B A B “A is a subset of B” A B if and only if every element of A is also an element of B. Examples: A = {3, 9}, B = {5, 9, 1, 3}, A B U A B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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} U A B Chumki Sarkar Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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 B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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 B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
DIFFERENCE OF TWO SETS U A B 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 B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
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 22-02-2019
LAWS OF SET OPERATIONS Associative Laws Commutative Laws i) A⋃B = B⋃A, ii) A⋂B = B⋂A for any two sets A & B Associative Laws i) A⋃(B⋃C)=(A⋃B)⋃C, ii) A⋂(B ⋂C)=(A⋂B)⋂C for any 3 sets A,B & C Distributive Laws i) A⋃(B⋂C)=(A⋃B)⋂(A⋃C), ii) A⋂(B⋃C)=(A⋂B)⋃(A⋂C) for any 3 sets A,B & C De Morgan’s Laws (A⋃B)C =AC⋂BC ii) (A⋂B)C=AC⋃BC for any two sets A & B Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Commutative Laws = = Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Associative Laws (i) Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Associative Laws (ii) Chumki Sarkar From equation (i) & (ii) Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Associative Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Associative Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
Distributive Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
De Morgan’s Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
De Morgan’s Laws Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019
THANK YOU Chumki Sarkar Presented by: Chumki Sarkar, Maheshtala College 22-02-2019