Sets Jeemain - Guru
Sets Jeemain - Guru
Sets Jeemain - Guru
GURU
JEE-Mathematics
SETS
SET : A set is a collection of well defined objects which are distinct from each other
Set are generally denoted by capital letters A, B, C, .... etc. and the elements of the set by a, b, c ....
etc.
If a is an element of a set A, then we write a A and say a belongs to A.
If a does not belong to A then we write a A,
e.g. The collection of first five prime natural numbers is a set containing the elements 2, 3, 5, 7, 11.
p
= :p, q I , q 0
q
R = the set of all real numbers.
R–Q = The set of all irrational numbers
( i ) Roster Method : In this method a set is described by listing elements, separated by commas and
enclose then by curly brackets
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TYPES OF SETS :
Null set or Empty set : A set having no element in it is called an Empty set or a null set or void set
it is denoted by or { }
e.g. A = {x N : 5 < x < 6} =
A set consisting of at least one element is called a non-empty set or a non-void set.
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JEEMAIN.GURU
JEE-Mathematics
Illustration 1 :
The set A = [x : x R, x2 = 16 and 2x = 6] equal-
(1) (2) [14, 3, 4] (3) [3] (4) [4]
Solution :
x2 = 16 x = ±4
2x = 6 x = 3
There is no value of x which satisfies both the above equations.
Thus, A =
Hence (1) is the correct answer
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The symbol '''' stands for "implies"
Proper subset : If A is a subset of B and A B then A is a proper subset of B. and we write A B
Note-1 : Every set is a subset of itself i.e. A A for all A
Note-2 : Empty set is a subset of every set
Note-3 : Clearly N W Z Q R C
Note-4 : The total number of subsets of a finite set containing n elements is 2 n
Universal set : A set consisting of all possible elements which occur in the discussion is called a Universal
set and is denoted by U
Note : All sets are contained in the universal set
e.g. If A = {1, 2, 3}, B = {2, 4, 5, 6}, C = {1, 3, 5, 7} then U = {1, 2, 3, 4, 5, 6, 7} can be taken
as the Universal set.
Power set : Let A be any set. The set of all subsets of A is called power set of A and is denoted by
P(A)
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JEEMAIN.GURU
JEE-Mathematics
e.g. Let A = {1, 2} then P(A) = {, {1}, {2}, {1, 2}}
e.g. Let P() = {}
P(P()) = {, {}}
P(P(P()) = {, {}, {{}}, {, {}}
Note-1 : If A = then P(A) has one element
Note-2 : Power set of a given set is always non empty
Illustration 2 :
Two finite sets of have m and n elements respectively the total number of elements in power set of first set is
56 more thatn the total number of elements in power set of the second set find the value of m and n
respectively.
Solution :
Number of elements in power set of 1st set = 2m
Number of elements in power set of 2nd set = 2n
Given 2m = 2n + 56
2m – 2n = 56
2n(2m – n – 1) = 23(23 – 1)
n = 3 and m = 6
Do yourself - 1 :
(i) Write the following set in roaster form :
A = {x|x is a positive integer less than 10 and 2x – 1 is an odd number}
(ii) Write power set of set A = {, {}, 1}
Disjoint Sets :
IF A B = , then A, B are disjoint.
e.g. if A = {1, 2, 3}, B = {7, 8, 9} then A B =
Note : A A' = A, A' are disjoint.
Symmetric Difference of Set s :
A B = (A – B) (B – A)
(A')' = A
A B B' A'
If A a nd B are a ny t wo set s, t hen
(i) A – B = A B'
(ii) B – A = B A'
(iii) A – B = A A B =
(iv) (A – B) B = A B
(v) (A – B) B =
(vi) (A – B) (B – A) = (A B) – (A B)
Venn Diagrame :
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Clearly (A – B) (B – A) (A B) = A B
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JEEMAIN.GURU
JEE-Mathematics
(iv) n(A B) = No. of elements which belong to exactly one of A or B
= n((A – B) (B – A))
= n(A – B) + n(B – A) [ (A – B) and (B – A) are disjoint]
= n(A) – n(A B) + n(B) – n(A B)
= n(A) + n(B) – 2n(A B)
= n(A) + n(B) – 2n(A B)
(v) n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(A C) + n(A B C)
(vi) Number of elements in exactly two of the sets A, B, C
= n(A B) + n(B C) + n(C A) – 3n(A B C)
(vii) number of elements in exactly one of the sets A, B, C
= n(A) + n(B) + n(C) – 2n(A B) – 2n(B C) – 2n(A C) + 3n(A B C)
(viii) n(A' B') = n((A B)') = n(U) – n(A B)
(ix) n(A' B') = n((A B)') = n(U) – n(A B)
Illustration 4 :
In a group of 1000 people, there are 750 who can speak Hindi and 400 who can speak Bengali. How
many can speak Hindi only ?How many can spak Bengali ? How many can spak both Hindi and Bengali ?
Solution :
Let A and B be the sets of persons who can speak Hindi and Bengali respectively.
then n(A B) = 1000, n(A) = 750, n(B) = 400.
Number of persons whos can speak both Hindi and Bengali
= n(A B) = n(A) + n(B) – n(A B)
= 750 + 400 – 1000
Number of persons who can speak Hindi > = n(A – B) = n(A) – n(A B) = 750 – 150 = 600
Number of persons Whos can speak Bengali only
= n(B – A) = n(B) – n(A B) = 400 – 150 = 250
Do yourself - 2 :
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