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JEE-MathematicsSETS
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.
SOME IMPORTANT NUMBER SETS :
N = Set of all natural numbers
= {1, 2, 3, 4, ....}
W = Set of all whole numbers
= {0, 1, 2, 3, ....}
Z or I set of all integers
= {.... –3, –2, –1, 0, 1, 2, 3, ....}
Z+ = Set of all +ve integers
= {1, 2, 3, ....} = N.
Z– = Set of all –ve integers
= (–1, –2, –3, ....}
Z 0 = The set of all non-zero integers.
= {±1, ±2, ±3, ....}
Q = The set of all rational numbers.
p
= :p, q I , q 0
q
R = the set of all real numbers.
R–Q = The set of all irrational numbers
e.g. 2 , 3, 5 , .... , e, log2 etc. are all irrational numbers.
METHODS TO WRITE A SET :
( i ) Roster Method : In this method a set is described by listing elements, separated by commas and
enclose then by curly brackets
\\NODE6\E_NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\01-SET\1.THEORY
e.g. The set of vowels of English Alphabet may be described as a, e, i, o, u
( i i ) Set Builder Form : In this case we write down a property or rule p Which gives us all the element
of the set
A = {x : P(x)}
e.g. A = {x : x N and x = 2n for n N}
i.e. A = {2, 4, 6, ....}
e.g. B = {x2 : x z}
i.e. B = {0, 1, 4, 9, ....}
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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,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
Singleton : A set consisting of a single element is called a singleton set.
e.g. Then set {0}, is a singleton set
Finite Set : A set which has only finite number of elements is called a finite set.
e.g. A = {a, b, c}
Order of a fi nite set : The number of elements in a finite set is called the order of the set A and is
denoted O(A) or n(A). It is also called cardinal number of the set.
e.g. A = {a, b, c, d} n(A) = 4
Infinite set : A set which has an infinite number of elements is called an infinite set.
e.g. A = {1, 2, 3, 4, ....} is an infinite set
Equal sets : Two sets A and B are said to be equal if every element of A is a member of B, and every
element of B is a member of A.
If sets A and B are equal. We write A = B and A and B are not equal then A B
e.g. A = {1, 2, 6, 7} and B = {6, 1, 2, 7} A = B
Equivalent set s : Two fi nite set s A and B are equivalent if their number of element s are same
ie. n(A) = n(B)
e.g. A = {1, 3, 5, 7}, B = {a, b, c, d}
n(A) = 4 and n(B) = 4 n(A) = n(B)
Note : Equal set always equivalent but equivalent sets may not be equal
Subsets : Let A and B be two sets if every element of A is an element B, then A is called a subset of
B if A is a subset of B. we write A B
Example : A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6, 7} A B
\\NODE6\E_NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\01-SET\1.THEORY
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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, 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}
Some Operation on Sets :
(i) Union of two sets : A B = {x : x A or x B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} then A B = {1, 2, 3, 4}
(ii) Intersection of two sets : A B = {x : x A and x B}
e.g. A = {1, 2, 3, }, B = {2, 3, 4} then A B = {2, 3}
(iii) Difference of t wo set s : A – B = {x : x A and x B}
e.g. A = {1, 2, 3}, B = {2, 3, 4} ; A – B = {1}
\\NODE6\E_NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\01-SET\1.THEORY
(iv) Complement of a set : A' = {x : x A but x U} = U – A
e.g. U = {1, 2, ...., 10}, A = {1, 2, 3, 4, 5} then A' = {6, 7, 8, 9, 10}
(v) De-Morgan Laws : (A B)' = A' B' ; (A B)' = A' B'
(vi) A – (B C) = (A – B) (A – C) ; A – (B C) = (A – B) (A – C)
(vii) Distributive Laws : A (B C) = (A B) (A C) ; A (B C) = (A B) (A C)
(viii) Commutative Laws : A B = B A ; A B = B A
(ix) Associative Laws : (A B) C = A (B C) ; (A B) C = A (B C)
(x) A = ; A U = A
A = A ; A U = U
(xi) A B A ; A B B
(xii) A A B ; B A B
(xiii) A B A B = A
(xiv) A B A B = B
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Illustration 3 :
Let A = {x : x R, |x| < 1] ; B = [x : x R, |x – 1| 1] and A B = R – D, then the set D is-
(1) [x : 1 < x 2] (2) [x : 1 x < 2] (3) [x : 1 x 2] (4) none of these
Solution :
A = [x : x R, –1 < x < 1]
B = [ x : x R : x – 1 –1 or x – 1 1]
= [x : x R : x 0 or x 2]
A B = R – D
where D = [x : x R, 1 x < 2]
Thus (2) is the correct answer.
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 :
\\NODE6\E_NODE6 (E)\DATA\2014\KOTA\JEE-ADVANCED\SMP\MATHS\UNIT#02\ENG\PART-2\01-SET\1.THEORY
Clearly (A – B) (B – A) (A B) = A B
Note : A A' = , A A' = U
SOME IMPORTANT RESULTS ON NUMBER OF ELEMENTS IN SETS :
If A, B and C are finite sets, and U be the finite universal set, then
(i) n(A B) = n(A) + n(B) – n(A B)
(ii) n(A B) = n(A) + n(B) A, B are disjoint non-void sets.
(iii) n(A – B) = n(A) – n(A B) i.e. n(A – B) + n(A B) = n(A)
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