Let and be non-empty sets such that || > || .
If the number of elements in ∪ is atmost 42 and at least 22 (that is 22≤|∪|≤42), then what is the
value of || ?
Answers
Answer:
1.1 Overview
This chapter deals with the concept of a set, operations on sets.Concept of sets will be
useful in studying the relations and functions.
1.1.1 Set and their representations A set is a well-defined collection of objects.
There are two methods of representing a set
(i) Roaster or tabular form (ii) Set builder form
1.1.2 The empty set A set which does not contain any element is called the empty
set or the void set or null set and is denoted by { } or φ.
1.1.3 Finite and infinite sets A set which consists of a finite number of elements is
called a finite set otherwise, the set is called an infinite set.
1.1.4 Subsets A set A is said to be a subset of set B if every element of A is also an
element of B. In symbols we write A ⊂ B if a ∈ A ⇒ a ∈ B.
We denote set of real numbers by R
set of natural numbers by N
set of integers by Z
set of rational numbers by Q
set of irrational numbers by T
We observe that
N ⊂ Z ⊂ Q ⊂ R,
T ⊂ R, Q ⊄ T, N ⊄ T
1.1.5 Equal sets Given two sets A and B, if every elements of A is also an element of
B and if every element of B is also an element of A, then the sets A and B are said to
be equal. The two equal sets will have exactly the same elements.
1.1.6 Intervals as subsets of R Let a, b ∈ R and a < b. Then
(a) An open interval denoted by (a, b) is the set of real numbers {x : a < x < b}
(b) A closed interval denoted by [a, b] is the set of real numbers {x : a ≤ x ≤ b)
2 EXEMPLAR PROBLEMS – MATHEMATICS
(c) Intervals closed at one end and open at the other are given by
[a, b) = {x : a ≤ x < b}
(a, b] = {x : a < x ≤ b}
1.1.7 Power set The collection of all subsets of a set A is called the power set of A.
It is denoted by P(A). If the number of elements in A = n , i.e., n(A) = n, then the
number of elements in P(A) = 2n
.
1.1.8 Universal set This is a basic set; in a particular context whose elements and
subsets are relevant to that particular context. For example, for the set of vowels in
English alphabet, the universal set can be the set of all alphabets in English. Universal
set is denoted by U.
1.1.9 Venn diagrams Venn Diagrams are the
diagrams which represent the relationship between
sets. For example, the set of natural numbers is a
subset of set of whole numbers which is a subset of
integers. We can represent this relationship through
Venn diagram in the following way.
1.1.10 Operations on sets
Union of Sets : The union of any two given sets A and B is the set C which consists
of all those elements which are either in A or in B. In symbols, we write
C = A ∪ B = {x | x ∈A or x ∈B}
Fig 1.1
Fig 1.2 (a) Fig 1.2 (b)
Some properties of the operation of union.
(i) A ∪ B = B ∪ A (ii) (A ∪ B) ∪ C = A ∪ (B ∪ C)
(iii) A ∪ φ = A (iv) A ∪ A = A
(v) U ∪ A = U
Intersection of sets: The intersection of two sets A and B is the set which
consists of all those elements which belong to both A and B. Symbolically, we
write A ∩ B = {x : x ∈ A and x ∈ B}.
When A ∩ B = φ, then A and B are called disjoint sets.
Fig 1.3 (a) Fig 1.3 (b)
Some properties of the operation of intersection
(i) A ∩ B = B ∩ A (ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)
(iii) φ ∩ A = φ ; U ∩ A = A (iv) A ∩ A = A
(v) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
(vi) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
Difference of sets The difference of two sets A and B, denoted by A – B is defined
as set of elements which belong to A but not to B. We write
A – B = {x : x ∈ A and x ∉ B}
also, B – A = { x : x ∈ B and x ∉A}
Complement of a set Let U be the universal set and A a subset of U. Then the
complement of A is the set of all elements of U which are not the elements of A.
Symbolically, we write
A′ = {x : x ∈ U and x ∉ A}. Also A′ = U – A
Some properties of complement of sets
(i) Law of complements:
(a) A ∪ A′ = U (b) A ∩ A′ = φ
(ii) De Morgan’s law
(a) (A ∪ B)′ = A′ ∩ B′ (b) (A ∩ B)′ = A′ ∪ B′
(iii) (A′ )′ = A
(iv) U′ = φ and φ′ = U