Set is a basic and unifying idea of mathematics. In fact all mathematical ideas can be expressed in terms of sets. In almost whole of the business mathematics the set theory is applied in one form or the other.
Definition: A set is a collection of well defined and different and different objects.
By the words 'well defined' we mean that we are given a rule with the help of which we can say whether a particular object belongs to the set or not . The word 'different' implies that repetition of objects is not allowed.
Element of a Set
Each object of the set is called an element of the set.
Examples of sets
(i) The set of days of a week .
(ii) The set of integers form 1 to 10000.
(iii) The set of even integers.
Methods of Designating a set
a set can be specified in two ways:
(1) Tabular , Roster or Enumeration Method :
When we represent a set by listing all its elements within braces i.e curly brackets { } , separated by commas, it is called the tabular , roaster or enumeration method.
(i) A set of vowels: A={ a, e, i, o, u }
(ii) A set of odd natural numbers: C={1,3,5,.............}.
(2) Sector, Set-builder, Rule Method or Predicate Form:
In this method , we do not list all the elements but the set is represented by specifying the defining property .
For example ,
A={ x: x is a vowel in English alphabets }
B= {x: x is a positives even integer up to 10 }
ILLUSTRATIVE EXAMPLES
Examples1. Which of the following are sets? justify your answer .
(i) The collection of all months of a year beginning with letter J
ans: It is a set as January, June, July, are three months of a year staring with J.
(ii) The collection of ten talented writers of India
ans: It is not a set as concept of ten talented writers of India is vague.
(iii) A team of eleven best cricket batsmen of the world
ans: It is not set as concept of eleven best cricket batsmen of the world is vague.
(iv) The collection of all boys in your class
ans: It is a set as collection of all boys in a class is well defined and of different boys.
(v) The collection of all natural numbers less than 100
ans: It is a set as natural numbers less than 100 are 1,2........99.
(vi) The collection of novels written by the writer prem cahnd
ans: It is a set as collection of novels written by the writer prem chand is well defined and of different objects.
(vii) The collection of all even integers
ans: It is a set as collection of all even integers is well defined.
(viii) The collection of different questions in this chapter
ans: It is a set as collection of different questions of a chapter is well defined.
(ix) A collection of most dangerous animal of the world
ans: It is a not a set as concept of most dangerous animal is vague.
Example: Write the following sets in Roster form :
(i) A={x ∈ N: x²=25}
solution: ∴ x²=25
∴ =5,-5
∴ But x ∈ N, ∴ x=5
∴ A={5}
(ii) |x|=x as x ∈ N
∴ |x| ≤ 4
∴ x≤4
∴ x=1,2,3,4
∴ B={1,2,3,4}
(iii) D={x:x is a positive multiple of 3 and 7 but less than 28}
∴ since x is a positive multiple of 3 and 7
∴ x is a multiple of 21
∴ D={21}
EXERCISE 1 (a)
1.What is the difference between a collection and set ? Give reason to support your answer.
ans: Every set is a collection but every collection is not necessary a set . Only a collection of well defined and different objects is a set.
2. Which of the following collection are set?
(i) The collection of intelligent student in india.
Ans: Not a set
(ii) The collection of positive multiples of 5.
Ans: Set
3. Write the set of all vowels in the English alphabet, which precede q.
Ans: {a, e, i, o}
4. List all the elements of the following sets:
(I) A ={x : x is an odd natural number}
Ans: = {1,3,5,7,........}
(II) B={x : x is an integer, -1/2<x<9/2}
Ans:= {0,1,2,3,4}
(III) C={x : x is an integer, x² ≤ 4}
Ans: = {-2,-1,0,1,2}
(IV) D= {x : x is a month of a year not having 31 days }
Ans: ={Feb,apr,june,sep,nov}
(V). E= {x : x is a letter in the word "LOYAL"}
Ans: = {L,OY,A}
(VI) F= {x : x is a consonant in the English alphabet which precedes k}
ANS: {b,c,d,f,g,h,j}
5. Write the solution set of the equation x²+x-2=0 in roaster form.
Ans:
x²+x-2=0
x²+2x-1x-2=0
x(x+2)-1(x+2)=0
x-1=0 ; x+2=0
x=1 ; x=-2
6. write the following sets in Roaster form:
(I) {x : x is a vowel before g in the English alphabet}
Ans: {a,e}
(II) {x ∈ N : x is a prime number between 6 and 30}
Ans: {7,11,13,17,19,23,29}
(III) {x ∈ N : 3x+5<31}
Ans: {1,2,3,4,5,6,7,8}
(IV) { x: x²+5x+6=0}
Ans: {-2,-3}
7. Write the set A = { 1,4,9,16,25,.......} in the set - builder form
Ans: A={x: x is the square of natural numbers}
8. Write the following sets in the set builder form:
(I) {3,6,9,12}
Ans:{x: x =3 n and 1 ≤ n ≤ 4 where n ∈ N}
(II) {2,4,8,16,32}
Ans: {x: x = 2n and 1 ≤ n ≤ 5 where n ∈ N}
(III) {5,25,125,625}
Ans: {x: x = 5n and 1 ≤ n ≤ 4 where n ∈ N}
(IV) {2,4,6,......}
Ans: {x: x is an even natural number}
(V) {1,4,9,......,100}
Ans: {x: x = n² and 1 ≤ n ≤ 10 where n ∈ N}
9.Write the set of all real numbers which can not be written as the quotient of two integers in the set builder form.
Ans: {x: x is real and irrational }
10. Write the set {1/2, 2/3 3/4, 4/5, 5/6, 6/7}
Ans: {x: x=n/n+1, n is a natural number and 1 ≤ n≤ 6}
11. Describe the following sets by set property method.
(I) A={1, 1/2, 1/3, 1/4, 1/5,........}
Ans: A={x: x=1/n, n ∈ N}
(II) A={1/4, 1/8, 1/16, 1/32, 1/64,1/128}
Ans: A={x: x=1/2n, n ∈ N and 2≤ n≤ 7}
(III) A={1/3, 1/6, 1/9, 1/12, 1/15, 1/18}
Ans: A= {x: x=1/3n , n ∈ N and 1≤ n≤ 6}
Art-3 Type of Sets
Finite Set
A set is said to be finite if it has finite number of element.
Examples:
A={2,4,6,8}
B={x: x is a student of D.A.V. College Amritsar}
C={x: x is citizen of India}
D={x: x ∈ N and 5<x<b}
Infinite Set
A set is said to be infinite if it has an infinite number of elements.
Examples:
A={1,2,3,........}
B={x: x is an odd integer}
C={x: x is a multiple of 6}
Singleton Set
A set containing only one element is called singleton or a unit set.
Examples:
A={x: x is a perfect square and 30≤ x≤ 40} = {6}
B={x: x is a positive integer satisfying x²=4} {2}
Empty ,Null or Valid Set:
A set which contains no element , is called a null set and is denoted by Φ (read as phi)
in roaster method , Φ is denoted by { }
Examples:
A={x: x is a positive integer satisfying x²=1/4}
B={x: x is a fraction satisfying x²=9}
C={x: x is an even prime number greater than 2}
A set which is not empty i.e. contains at least one element is called non - empty set.
Order or Cardinal Number of a Finite Set
The number of different elements of a finite set A is called the order of A and is denoted by O(A).
Examples. If A={2,3,6,8}, then O(A)=4
Equivalent Sets
Two finite sets A and B are said to be equivalent sets if the total number of elements in A is equal to the total number of element in B.
In other words two finite sets are equivalent if their cardinal numbers are same .
Example: Let A={1,2,3,4,6}, B={1,2,7,9,12}
∴ O(A) = 5 =O(B) => A and B are equivalent sets.
we write the above fact as A~B.
Equality of sets
two sets A and B are said to be equal if both have the same elements. In other words, two sets A and B are equal when every element of A is an element of B and every element of B is element of A.
Example: A={1,2,3,4,5,6,7,8,9,10}
B={x: x is a natural number and 1 ≤ x ≤ 10}
Here A=B.
ILLUSTRATIVE EXAMPLES
Examples 1. Which of the following sets are null sets:
(i) A={x :x < 1 and x>3}
sol: There is no number which is less than 1 but greater than 3
∴ given set is a null set
∴ A=Φ
(ii) B={x : x²=9 and 3x=7}
x²=9 => x= -3, 3
3x = 7 => x=7/3
∴ there is no x which satisfies both the equation x²=9 and 3x=7
∴ given set is a null set.
∴ B=Φ
(iii)C={x: x²-1 =0, x ∈ R}
x²-1=0 =>x²=1 => x=-1, 1 ∈ R
∴ C= {-1, 1}
∴ C is not a null set i.e. C≠Φ
(iv) D={x: |x| =1,x ∈ Z}
Now |x| =1 => x=-1,1 ∈ Z
∴ D={-1, 1}
∴ D is not a null set i.e. D=Φ
Examples 2. State which of the following sets are finite or infinite :
(i) {x : x ∈ N and (x-1) (x-2) =0}
sol. Let (x-1) (x-2)=0 => x=1,2 ∈ N
∴ ={1,1}
∴ A is finite.
(ii) {x: x ∈ N and x² =4}
sol. x²=4 =>x²-4=0 =>(x-2) (x+2)=0
∴ x=-2,2
∴ x=2 as -2 ∉ N
∴ A is finite.
(iii) {x: x ∈ N and 2x-1=0}
sol. = 2x-1=0 => x=1/2 ∉ N
=Φ
∴ A is finite.
(iv) {x: x ∈ N and x is prime}
sol. let A {2,3,5,7,11,13,17,........}
∴ A is infinite
(v) {x: x ∈ N and x is odd}
sol. {1,3,5,7,9,............}
A is infinite.
Example 3. Are the following pair of sets equal ? Give reason
(i) A={2,3}
B={x: x is a solution of x²=5x=6=0}
sol. ={-2,-3} [ x²+5x+6=0 => (x+2) (x+3)=0
=> -2,-3]
∴ A≠B as 2 ∈ A but 2 ∉ B.
(ii) A={x : x is a letter in the word FOLLOW}
B={y: y is a letter in the word WOLF}
sol. A={x : x is a letter in the word FOLLOW}
= {F,O,L,L,O,W} = {F,O,L,W}
B={y: y is a letter in the word WOLF}
= {W,O,L,F}
Since every letter of A is in B and every letter of B is in A.
∴ A=B.
Example : 4. Which of the following sets are equal?
A={1,2}
B={1,2,1,2,1,1}
C={x: x²-3x+2=0}.
sol. A={1,2}
B={1,2,1,2,1,1}=A={1,2}
[ 1,2 are only different element of B ]
Now x²-3x+2=0 => (x-1) (x-2)=0 =>x=1,2
C={1,2}
A=B=C.
Example 5. Is the Set
A={x : x3=8 and 2x+3=0} empty ? justify.
sol. A={x x3=8 and 2x+3=0}
Now x3=8 => x3-8=0
x3-(2)3 =0
(x-2)(x²+2x+4)=0
Also 2x+3=0 => x=3/2
there is no value of x which satisfies (x)3 =8 and 2x+3=0
A is an empty set.
Nice 2
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