W C EL E M O 1 Sets

W C EL E M O

1. Sets Name – Sub. - Algebra(p. p. t. ) Std. -9 th Div. -A Smt: Nimbore v. p. madam School name: P. Dr. V. V. Patil Vidyalaya Loni. Head Master: Game Sir.

1. 1]Defination of Set – A well defined collection of objects is called a set. 1. 2] Methods of a writing sets – a) Listing Method b) Rule Method a) Listing Method While writing a set by the Listing method , we proceed as follows : i]We write the given names of the set we put= sign and write all its elements enclose within braces {} ii]Elements are separated by commas. iii]An element, even if repeated , is listed only once. Ex. The set of digits in the number 2332= {2, 3} iv]The order of the elements in a set is immaterial. Ex. The set of all days of a week = A= {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

b] Rule Method. In this method, we describe the elements of the set by specifying the property which determines the elements of the set Uniquely. Ex. The set of prime numbers from 1 to 25 - A= {2, 3, 5, 7, 11, 13, 17, 19, 23, } this can be written as in set builder form as: A= {x/x is a prime number less than 25 } 1. 3]Venn Diagram : A set is a represented by any closed figure such as circle, rectangle, Triangle etc. The Diagrams representing sets are called venn diagrams. Ex. Set A= {4, 6, 9, } B = {1, 5, 8, 13} C={ a, b, c, d, e} A B E. 6 . 4. 9 . 13 . 5. 8 . a. b. c. d. e

1. 4]Types of a sets : Sr. No. Sets Characteristics of a Sets Types of a sets Empty set or null set And is denoted by O 1. a) b) { } {x/x is even prime number greater than 2} Set having no elements Set having no element 2. a) {2} Set having a only one element Singleton Set Having a only one element b) {y/y -5 =0} 3. a) b) {1, 2, 3, 4, 5, 6} {x/x is a book on your school library } Counting of element terminates at a certain stage. Finite set 4. a) b) {101, 102, 103, ……. } {p/p is a natural number} Counting elements do not terminate at any stage Infinite set

1. 5]Subset: i]. 5. 6. 1. 7. 2. 3. 4 In the venn diagram, A={1, 2, 3, 4, 5, 6, 7} and B ={5, 6, 7} Every element of the set B is an element of the set A. Here, the set B is a subset of set A. ii] Definition: i] if every element of set B is an element of Set A, then set B is said to be the subset of set A. ii] if ‘B is a subset of A’, we write it as B iii]if ‘B is a not subset of A’, we write it as B

Proper subset: Let, p={b, a} and Q. ={a, b, c} Here, P is a subset of a Q. The set Q contains at least one element which is not in the set P. Hence, P is a proper subset of Q. It is denoted as P The set Q is said to be super set of the set P. It is denoted as Q [Note: i] Every set is a subset of itself. Ex. A ii] Empty set is subset of every set. Ex.

1. 6] Universal set: Definition: A suitably chosen non empty set, of which all the sets under consideration are the subsets of that set, is called the Universal set. Ex : A={X X is a student in IX A class of your school} B= { Y Y is a student in IX B class of your school} C={Z Z is a student in IX C class of your school} U={u u is a student of standard IX of your school} A U, B U and, C U

1. 7] Operations on sets: a] Equality in sets: Two sets A and B are said to be equal, if they contain exactly the same elements. We write this as A=B If sets A and B are not equal, we write this as A B. Ex: i] A={ X X is a letter in the word ‘tea’} B={ Y Y is a letter in the word ‘eat’} The sets A and B contain the same elements therfore, A=B

[b] Intersection of sets: The intersection of two sets A and B is the set of common elements of A and B. we write such as set A B and read as ‘A intersection B’. Ex: i] Let, A={2, 4, 6, 8, 10, 12} B={3, 6, 9, 12, 18} Ans: A B={6, 12} [ Common is(6, 12)] E Ex: ii] E={2, 3, 4, 5, 6}, F={3, 4, 5, } Ans: E F ={3, 4, 5}= F. F figure of ex. ii] . 3. 2. 4. 5 . . 6

Disjoint set: If the two sets have no common element, then the two sets are said to be disjoint sets. Ex: A={2, 5, 7, 9} and B={1, 4, 6, 8}. Here, A B= Hence, the sets A & B are disjoint sets. C] Union of sets: The union of two sets A & B is a set which consists of all the elements of set A and all the elements of set B. we write such a set as A B and read it as ‘A union B’.

Ex: i] U P={1, 3, 5}, Q ={2, 4, }. 1 Q ={1, 2, 3, 4, 5}. 3. 5 [ P and Q are disjoint sets] Properties of union sets: Let, A & B be the subsets of the universal set U. then, i] A B =B A. . . . [Commutative property] ii] A [B C] =[A B] C ……[Associative Property] P Q. 2 Hence, P. 4

Distributive Property: i] A [B C]=[A B] [A C]. ii] A [B C]=[A B] [A C] e] Complement of a set: Consider , U= {X X is a natural number, x< 11} A= {2, 5, 7, 9} Write U in the roster form. U= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} U A. Write U- A. 2, . 7 U- A= {1, 3, 4, 6, 8, 10} , . 9, . 5

Observe the set U-A. it contains all those elements of U which are not in A. The set U-A is complement of the set A. Let A be the subset and U be the universal set, the set of all elements in U which are not in A i. S called the complement of A. It is denoted by A’ or A Note: i] A A’= ii] A A’=U

Elements in the sets: Let, A be the Finite set. The number of elements in the set A is denoted by, n(A). Ex, i] A={3, 5, 9}, n(A)=3. ii] B={5, 7, 9, 11}. n (B)=4. by using formula, n (A B)=n (A) +n (B)- n (A B) Ex. Let A and B be two sets such that n (A) =5, n (A B)=9, n (A B)=2, Find n (B). Ans; n (A B)= n (A)+n (B)- n (A B) …(Formula)

therfore 9 =5 + n (B) – 2 (Substituting the given values) therfore n(B) = 9 - 5+ 2 therfore n ( B )= 6.

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