SET THEORY
1) Set: A set is a collection of well defined and well distinguished objects of our perception or thought.
The words 'well defined objects ' imply that we must be given a rule with the help of which we should readily be able to say whether a particular object 'belongs to' the set of not. The words 'well distinguished objects ' imply that if the objects of the collection be named, then in doing so, the number of objects will not increase.
The set are usually denoted by capital letters of English alphabet viz, A, B, CA
2) ELEMENTS
The objects, which constitute the set, are said to be elements of the set.
These are also known as members or points of the set. The elements are usually denoted by small letters of English alphabet viz, a, b, c,....
i) If a is an element of the set A, we write it as a ∈A and is ready as " a belongs to A"
ii) If a is not an element of the set A, we write it as a∉ A and is ready as " a does not belong to A"
3) REPRESENTATION OF SETS
There are two methods to represent a set.
a) Roster /Tabulation Method. In this method, the set is represented by listing all its elements, saparating the elements by commas and enclosing them in curvilinear brackets.
b) Defining Property Method. In this method, the set is represented by specifying the common property of the elements.
Thus the set A is represented by A= {a: P(A) is true}.
Here 'a' stands for 'an arbitrary elements' of the set' and (:) stands for 'such that' and P(A) stands for 'common property '
4) FINITE AND INFINITE SETS
a) finite set. If it has finite number of elements.
b) Infinite set. If it has an infinite number of elements.
Order of a finite set is the number of elements it contains.
The order of a finite set A is denoted by O(A).
5) EMPTY SET
A set having no element is....
It is also called Null set or Void set.
6) SINGLETON SET
A set having only one element is....
7) SUB-SETS
a) Subset. Let A and B be two sets. Then the set A is said to be a subset of the set B if each element of A is also an element of B.
Symbolically, we write it as A⊆ B.
Here B is superset of A and is written as B⊇A.
b) Proper Subset. A set A is a proper subset of B if and only if each element of A is in B and there is atleast one element in B, which is not in A.
Symbolically, if A is a proper subset of B, then A ⊂B and A≠ B or A ⊂ B
8) COMPARABLE SETS
Two sets are said to be comparable iff either A ⊂B or B⊃A.
9) EQUAL AND EQUIVALENT SETS
a) Equal Sets. Two sets A and B are said to be equal (written as A= B) iff A ⊂B and B ⊂A.
Two sets A and B are said to be equal if they have exactly same elements.
b) Equivalent Sets , Two sets are said to be equivalent if they have same number of elements.
10) FAMILY OF SETS
A set said to be family of sets if it's elements are also sets.
This is also known as set of sets.
If A= {a,b}, then S={φ, {a}, {b}, {a,b} is the set of sets.
11) POWER SET
The set of all possible subsets of a set A is said to be the power set of A and is denoted by P(A).
If a= P{a,b,c}, then P(A)={φ, {a}{b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c}}.
12) UNIVERSAL SET
The main set under discussion or the set containing all possible values in the given frame of reference is said to be universal set and is denoted by U or E or X.
13) OPERATIONS IN SETS
a) Union of Sets.
(I) Let A and B be two sets. The union of A and B (denoted by A U B) is the set of all those elements which are either in A or in B or in both.
Symbolically, A∪B= x: x ∈A or x ∈ B}.
II) Let A₁, A₂,..., Aₙ be n(≥2) sets. Then the union of these (denoted by ⁿᵢ₌₁U Aᵢ) is the set of all those elements which are in Aᵢ (1≤ i≤ n) for atleast one value of i.
b) Intersection of sets.
I) Let A and B be two sets. The intersection of A and B (denoted by A∩B) is the set of all those elements which are in both A and B.
II) Let A₁, A₂, .....,Aₙ be n(≥2) sets. Then the intersection of these (denoted by ⁿᵢ₌₁ ∩Aᵢ) is the set of all those elements which are in Aᵢ(1≤ i≤ n) for each i.
14) FUNDAMENTAL RESULTS
i) Identity Laws, A ∪φ, A ∩φ= φ
ii) Idempotent Laws. A ∪A= A, A ∩A = A
iii) Commutative Laws. A ∪ B= B ∪ A, A ∩ B= B∩ A
iv) Associative Laws. A∪(B ∪C)= (A∪B) ∪ C
A∩(B∩C)= (A∩B)∩C
c) Distributive Laws, A∪ +B∩C)= (A∪B)∩(A ∪ C)
A∩(B∪C)= (A∩B)∪(A∩C).
15) Disjoit Sets
If and only if they have no common element.
Let A and B be two sets
Here A ∩B= φ.
16) DIFFERENCE OF SETS
Let A and B be two sets. Then (A - B) is the set of those elements of the set A which are not in the set B.
Symbolically. A - B= {x: x ∈A and x ∉B}
Similarly, B - A ={x: x ∈B and x ∉A}
17) SYMMETRIC DIFFERENCE OF SETS
Let A and B be two sets. Then their symmetric difference is the union of the sets A- B and B- A. This is denoted by A ∆B.
Symbolically
A∆B ={x: x ∈A - B or x ∈B - A}
18) COMPLEMENT OF A SET
Let X be the universal set and A be any set. Then the complement of the set A is the set of all those elements of X, which are not in the set A.
This is denoted by Aᶜ or A' or X - A
Symbolically, Aᶜ={x: x∈X and x ∉A}.
19) FUNDAMENTAL RESULTS
a) Xᶜ= φ, φᶜ= X
b) (Aᶜ)ᶜ= A
c) If A ⊆ B, then B⊆Aᶜ
d) A U Aᶜ= X and A∩Aᶜ = φ
e) De-morgan's Laws. (AUB)ᶜ = Aᶜ∩ Bᶜ
(A∩ B)ᶜ= AᶜU Bᶜ
20) USE OF SETS IN PRACTICAL PROBLEMS
If A, B, C are finite sets of n elements each, then
a) n(AUB)= n(A)+ n(B); if A, B are disjoint
b) n(AUB)= n(A)+ n(B) - n(A∩B); if A, B are not disjoint
c) n(A∩Bᶜ)= n(A) - n(A∩B)
d) n(B∩Aᶜ)= n(B)- n(A∩B)
e) n(AUB)= n(A∩Bᶜ)+ n(B∩Aᶜ)+ n(A∩B).
f) n(AUBUC)= n(A)+ n(B)+ n(C) - n(A∩B) - n(B ∩C) - n(A∩C)+ n(A∩B∩C).
21) ORDERED PAIR
An ordered pair is a pair of entries in the specified order.
In the ordered pair (a,b), a is the first element and b the second element.
22) CARTESIAN PRODUCT/DIRECT PRODUCT OF SETS
a) The set of all ordered pairs of elements (a,b); a ∈A, b ∈ B is called Cartesian product of two sets A and B and is denoted by A x B
Symbolically, A x B={(a,b): a ∈A, b∈B}.
b) The Cartesian product of n(>2) sets A₁, A₂,......Aₙ is the set of all ordered n-triples (a₁, a₂, .......aₙᵢ), where aᵢ∈ Aᵢ (1≤ i≤ a) and is denoted by A₁x A₂ x......x Aₙ or ⁿᵢ₌₁ΠAᵢ.
Symbolically, ⁿᵢ₌₁ΠAᵢ= {(a₁, a₂,......,aₙ): aₙ ∈ Aₙ, 1≤ i≤ n}.
23) FUNDAMENTAL RESULTS
a) Ax B≠ B x A
b) A x φ= φ x = φ
c) n(Ax B)= n(B x A)= n(A) x n(B)
d) If A ⊆B, C ⊆ D, then A x C ⊆ B x D
e) n(A₁x A₂ x......x Aₙ)= n(A₁) x n(A₂)......xXₙ (Aₙ)
IN FOCUS
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φΠ ᵢ
φφ
ᶜᶜᶜᶜφ
∪∪∪∪
ⱼ∉∆
₁₂ₙᵢⱼ∉∆
⊆⊇⊂∩∪∀≺∩⊃⊄⊅¹²₁₂ₙⁿᵢ₌₁ᵢ₁₂₌₁ᵢ₁₂ₙᵢⱼ∉∆ₙⁿᵢ
1) Let N be the set of non-negative integers, I the set of integers Nₚ the set of non positive integers, E the set of even integers and P the set of prime numbers. Then
a) I - N= Nₚ
b) N ₚ∩ Nₚ = φ
c) E ∩C = φ
d) N ∆Nₚ= I - {0}
2) Let A and B be two sets, then (AUB)ᶜ U(Aᶜ∩B) equals
a) Aᶜ b) Bᶜ c) A d) none
3) If A and B are two sets , then A∩(AUB)ᶜ equals
a) A B) B C) φ d) none
4) The set (A∩Bᶜ)ᶜ U (B∩C) equals
a) Aᶜ U Bᶜ b) Aᶜ U Cᶜ c) Aᶜ U B U C d) none
5) Let U be the universal set and AU BU C= U Then [(A - B) U(B - C) U(C A)]ᶜ equals
a) AUBUC b) A∩B∩C c) AU(B∩C) d) A∩(B UC)
6) The set (AUBUC)∩(A∩Bᶜ∩Cᶜ)∩Cᶜ equals
a) A∩C b) B U Cᶜ c) B ∩Cᶜ d) none
7) If A and B are two sets, then A ∩(AUB) equal
a) A b) B c) Aᶜ d) Bᶜ
8) If A={1,3,5,7,9,11,13,15,17}, B={2,3,.....18} and N is the universal set, then Aᶜ U ((A U B)∩ Bᶜ is
a) A B) N c) B d) none
9) If A and B are disjoint nonempty sets, then A - (A - B) equals
a) A B) B C) φ d) A U B
10) Which of the following is empty set?
a) {x: x is a real number and x² -1=0}
b) {x: x is a real number and x² +1=0}
c) {x: x is a real number and x² -9 =0}
d) {x: x is a real number and x²= x +2}
11) Which of the following is a singleton set?
a) {x: |x|=5, x ∈ I}
b) {x: |x|=6, x ∈ N}
c) {x: x²=5, x ∈ N}
d) {x: x²+ 3x +2=0, x ∈ N}
12) Which of the following does not have a proper subset:
a) {x: x ∈ N, 4< x<5}
b) {x: x ∈ Q}
c) {x: x ∈ WpQ, 4< x<5} d) none
13) If A, B and C are any three sets, then A - (BUC) equals
a) (A-- B) U(A - C)
b) (A- B) ∩(A - C)
c) (A-- B) U C
d) (A - B) ∩ C
14) If A, B, C are three sets, then A∩(B U C) equals
a) (AUB) ∩(AUC)
b) (A∩B) U(A∩C)
c) (AUB) U (AU C) d) none
15) If Q={x: x= 1/y, where x ∈N} then:
a) 0∈Q b) I∈ Q c) 2∈ Q d) 2/3 ∈Q
16) If A={x: x∈I, -2≤ x ≤ 2}, B==x: x∈ I, 0≤ x ≤ 3}
C={x: x ∈N, 1≤ x ≤ 2}
D={(x,y): (x,y) ∈N x N, x+ y= 8}, then
a) n(BUC)= 5
b) n(D)= 6
c) n(AU(BUC)(= 5 d) none
17) If for α∈ N, αN= {αx: x ∈ N}, then the set 8N ∩6N is
a) 8N b) 12N c) 24N d) 48N
18) Let n(A)= 3 and n(B)= 6 and A B. Then the number of elements in A ∩B is
a) 3 b) 9 c) 6 d) none
19) Sets A and B have 3 and 6 elements respectively. What can be the minimum number of elements in AUB ?
a) 3 b) 6 c) 9 d) 18
20) Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of second set. The values of m and n are
a) 7,6 b) 6,3 c) 5,1 d) 8,7
21) If A, B and C are any three sets, then Ax (BUC) is
a) (AxB) U (Ax C)
b) (A UB) x (AU C)
c) (Ax B) ∩(Ax C)
d) none
22) If A, B and C are any three sets, then A x (B ∩C) is
a) (Ax B) U(Ax C)
b) (Ax B)∩(Ax C)
c) (AUB) x (AUC)
d) (A∩B) x (A∩C)
23) If S₁={1,2,3,.....20}, S₂= {a,b,c,d}, S₃={b,d,e,f}. The number of elements of (S₁x S₂) U(S₁x S₃) is
a) 100 b) 120 c) 100 d) 40
24) If A={1,2,3,6,11,18,21}, B={5,7,9} and N is the universal set, then Aᶜ U(A U B) ∩ Bᶜ equals
a) A b) B C) N d) N - A
25) The set (AUB)∩(A ∩Bᶜ∩Cᶜ)ᶜ ∩Cᶜ equals
a) A∩C b) B∩Cᶜ c) Bᶜ∩Cᶜ d) none
26) Suppose A₁, A₂, ......A₃₀ are thirty sets, each having 5 elements and B₁, B₂, .....Bₙ are n sets, each element of S belongs to exactly 10 if the Aᵢ's and exactly 9 of Bᵢ's. Then n is equal to
a) 15 b) 3 c) 45 d) none
27) Consider the set of all determinants of order 3 with entries 0 or 1 only. Let B be the subset of A consisting of all determinants with value 1. Let C be the subset of the set of all determinants with value -1. Then
a) C is empty
b) B has as many elements as C has
c) A= B U C
d) B has twice as many elements as C has
28) If (1,3),(2,5) and (3,3) are the elements of Ax B and the total number of elements in A x B is 6, then the remaining elements of Ax B are
a) (1,5),(2,3),(3,5)
b) (5,1),(3,2),(5,3)
c) (1,5),(2,3),(5,3) d) none
29) If A, B, C be three sets such that AUB = AUC and A∩B= A∩C, then
a) A= B b) B= c c) A= C d) A= B = C
30) Let A={x,y): y= eˣ, x∈ R}, B= [(x,y): y= e⁻ˣ, x ∈ R}. Then
a) A∩B= φ b) A∩B≠ φ c) AUB= R d) none
31) Let A={(x,y): y= eˣ, x ∈R}, B={(x,y): y= x, x∈ R}. Then
a) B⊆ A B) A ⊆ B C) A∩B= ∪φ d) AU B = A
32) If X={4ⁿ - 3n -1: n∈ N} and Y={9(n -1); n∈ N} then X U Y is
a) X b) Y c) N d) none
33) If X= {8ⁿ - 7n -1: n∈ N} and Y= {49(n -1)| n ∈ N}, thn
a) X⊂ Y b) Y⊂ X c) X= Y d) none
34) If the sets A and B are defined as:
A={(x,y): y= 1/x, 0 ≠ x R}
B={(x,y): y= - x, x ∈ R}, then:
a) A∩B= A b) A∩B= B c) A∩B= φ d) none
35) If A={φ, {φ}}, then the power set of A is
a) A B) {φ, {φ}, A} c) {φ, {φ}, {{φ}}A} d) none
36) In a class of 100 students, 55 students have passed in mathematics and 67 students have passed in Physics. Then the number of students who have passed in Physics only is
a) 22 b) 33 c) 10 d) 45
37) Out of 800 boys in a school, 224 played cricket, 240 played hockey and 336 played basketball. Of the total, 64 played both basketball and hockey, 80 played cricket and hockey; 24 played all the three games. The number of boys which did not play any game is
a) 128 b) 216 c) 240 d) 160
38) From 50 students taking examinations in mathematics, physics and chemistry, 37 passed mathematics, 24 physics and 43 chemistry. Atmost 19 passed mathematics and physics, atmost 29 mathematics and chemistry and atmost 20 physics and chemistry. The largest possible number that could have passed all three examination is
a) 9 b) 10 c) 12 d) none
39) Of the members of three athletic teams in a school 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is
a) 43 b) 49 c) 76 d) none
40) In a college of 300 students, every student reads 5 newspaper and every newspaper is ready by 60 students. The number of newspaper is
a) atleast 30
b) atmost 20
c) exactly 25 d) none
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