(є:belongs to , є:does not belongs to) A well defined collection of objects is called a set. Individual object in the set is called an element or member of the set. Sets are denoted by capital alphabets e.g. A,B,C, etc.The elements of sets are generally denoted by small alphabets e.g. a,b,c etc. If x is an element of the set X then we write it as xє X and if x is not an element of set X then we write x є X. (є:belongs to , є:does not belongs to)

Common notations (a) N= the set of nonnegative integers or natural number={1,2,3,...} (b) W= the set of whole numbers= {0,1,2,3,...} (c) I=the set of integers={…,-3,-2,-1,0,1,2,3,...} (d) Q=the set of rational numbers (e) Q+=the set of positive rational numbers (f) R=the set of real numbers

Sets Methods of writing sets (a) Listing method(Roster form) (b)Rule method(Set builder form)

Sets LISTING METHOD(Roster form) In this method ,first write the name of the set , put is equal sign and write all its elements enclosed within curly brackets{ }. Elements are separated by commas. An element , even if repeated , is listed only once. The order of the elements in a set is immaterial.

e .g. 1) The set of all natural numbers less than ten. 2) The set of colours in the rainbow. B={red ,orange ,yellow ,green,blue,indigo,voilet} 3) C =The set of letters in the word ‘MATHEMATICS’ C={m , a , t ,h ,i ,c , s}

Rule Method(Set builder form) In the set builder form we describe the elements of the set by specifying the property or rule that uniquely determines the elements of the set. Consider the set P={1,4,9,16,} P={x x =n2,nєN , n=1,2,3,4,5} In this notation , the curly bracket stands for ‘the set of ’, vertical line stands for, such that. Here ‘x’ represents each elements of that set. And read as “P is the set of all x such that x is equal to n2,where n є N and n is less than or equal to 5.”

Venn-Diagrams : A A A A or .4 .5 .1 .4 .2 .1 .3 .1 .4 .3 .1 .3 or or Many ideas or concepts are better understood with help of diagrams . Such presentation used for sets is called Venn-diagram .For this use the closed figure and elements of the sets represented by points in that closed figure. A={ 1,2,3,4,5} can be represented as A A A A or .4 .5 .1 .4 .2 .1 .3 .1 .4 .3 .1 .3 or or .2 .5 .2 .4 .2 .5 .3 .5

Sets 1) Empty set or Null set : 2) Singleton Set : Types of sets: A set which does not contain any element is called Empty or Null set. It is denoted by {} or Φ e.g. The set of men whose heights are more than 5meter. 2) Singleton Set : A set containing exactly one element is called a Singleton set. e.g. 1)P={x:x is a natural number,4<=x<=6} 2)E={0}

Sets Infinite set : Finite set : If the counting process of elements of a set terminates , such a set is called a finite set. e.g. B={1,2,3,…,200000} D={a,e,i,o,u} Infinite set : If the counting process of elements of a set do not terminates at any stage , such a set is called a Infinite set. e.g. N={1,2,3,4,…}

Sets If B is a subset of A and the set A contains at least one element which is not in the set B, then the set B is the Proper subset of set A. It is denoted as B U A. In this case the set A is said to be the Super set of set of the set B and is denoted as B U A. Note:1) Every set is a subset of itself. 2)Every set is a subset of every set.

Sets Universal set : A suitable chosen non-empty set of which all the sets under consideration are the subsets of that set is called the Universal set. e.g. If A={2,3},B={1,4,5},C={2,4} then U={1,2,3,4,5} can be taken as the universal set of the sets A,B and C.

Operations on sets Sets (a)Equality: If A is subset of B and B is subset of A, then A and B are said to be equal sets and are denoted by A=B. e.g. If A={2,4,6,8},B={4,8,2,6,} then A=B.

(b)Intersection of sets: If A={1,2,3,4, 6,7} and B={2,4,5,6,8} then Operations on sets : (b)Intersection of sets: If A={1,2,3,4, 6,7} and B={2,4,5,6,8} then C={2,4,6} is called the intersection of the sets A and B. The set of all common elements of A and B is called the intersection A and B. A U B B A .5 .1 .2 .3 .4 .8 .6 .7

Here both sets A and B have no common elements . Operations on sets : Disjoint sets: Let A={1,2,3,4} and B={5,6,7,8} Here both sets A and B have no common elements . Therefore set A and B are Disjoint sets. A ∩ B={ } or Φ A .1 B .3 .4 .5 .6 .7 .2 .8

Properties of Intersection of sets: 1)A ∩ B =B∩ A (commutative property) 2)A ∩ (B ∩ C) =(A ∩ B) ∩ C (associative property) 3) A ∩ B ⊆ A; A ∩ B ⊆ B 4)If A ⊆ P; B ⊆ P then A ∩ B ⊆ P 5)If A ⊆ B then A ∩ B=A. If B ⊆ A then A ∩ B = B 6)A ∩ Φ = Φ and A ∩ A =A

A U B Let A={1,2,3,4} and B={4,5,6,1,8} be the sets. Operations on sets: (c) Union of sets: Let A={1,2,3,4} and B={4,5,6,1,8} be the sets. If we write set C , which contains all the elements of A and B together is called the Union of sets A and B.As follows C={1,2,3,4,5,6,8} .3 B A .4 .5 .6 .1 .2 .8 A U B

Properties of Union of sets: 1)A U B=B U A 2)A U (B U C)=(A U B) U C 3)A ⊆ (A U B) and B ⊆ ( A U B) 4)If A ⊆ B then (A U B) =B and (B U A) =A 5)(A U ø ) =A 6)(A U A)=A

B A .1 .6 .5 .3 .7 .8 .2 .4 A-B Operations on sets : (d)Difference of two sets : Consider the following two sets. A={1,2,3,4,5}and B={1,2,6,7,8} If we write the set C , which contains all the elements in set A but not in set B is called the Difference of sets A and B .As C={3,4,5} B A .6 .5 .1 .3 .7 .2 .8 .4 A-B

Properties of Difference of sets: 1)A - B ≠ B - A 2)A-B ⊆ A 3)If A ⊆ B, then A –B= ø 4)If A ∩ B= ø, then A - B =A

A={2,3,5} U A .1 .6 .2 .3 .4 .5 .7 .8 Operations on sets : (e)Complement of set : Consider U={ x:x is a natural number , x<9} A={2,3,5} First we U in the roster form U={1,2,3,4,5,6,7,8} then U-A ={1,4,6,7,8} Now if we observe (U-A). It contains all those elements of U which are not in A. Here, (U-A) is called the complement of A . It is denoted by A, or Ac . U A .1 .6 .2 .3 .4 .5 .7 (U-A) or Ac .8