SET LANGUAGE
Set
A set is a well-defined collection of objects.
Here “well-defined collection of objects” means that given a specific object it must be possible for us to
decide whether the object is an element of the given collection or not.
The objects of a set are called its members or elements.
For example,
1. The collection of all books in a District Central Library.
2. The collection of all colours in a rainbow.
3. The collection of prime numbers.
Notation
A set is usually denoted by capital letters of the English Alphabets A, B, P, Q, X, Y, etc.
The elements of a set is written within curly brackets “{ }”
If x is an element of a set A or x belongs to A, we write x ∈ A.
If x is not an element of a set A or x does not belongs to A, we write x ∉ A.
For example,
Consider the set A = {2,3,5,7} then
2 is an element of A; we write 2 ∈ A
6 is not an element of A; we write 6 ∉ A
Representation of a Set
The collection of odd numbers can be described in many ways:
(1) “The set of odd numbers” is a fine description, we understand it well.
(2) It can be written as {1, 3, 5, ...}
(3) Also, it can be said as the collection of all numbers x where x is an odd number.
All of them are equivalent and useful. For instance, the two descriptions “The collection of all
solutions to the equation x–5 = 3” and {8} refer to the same set.
A set can be represented in any one of the following three ways or forms:
Descriptive Form
In descriptive form, a set is described in words.
For example,
(i) The set of all vowels in English alphabets.
(ii) The set of whole numbers.
Set Builder Form or Rule Form
In set builder form, all the elements are described by a rule.
For example,
(i) A = {x : x is a vowel in English alphabets}
(ii) B = {x|x is a whole number}
Roster Form or Tabular Form
A set can be described by listing all the elements of the set.
For example,
(i) A = {a, e, i, o, u}
(ii) B = {0,1,2,3,...}
Types of Sets
Empty Set or Null Set
A set consisting of no element is called the empty set or null set or void set.
It is denoted by Q or { }.
For example,
(i) A={x : x is an odd integer and divisible by 2}
` A={ } or Q
(ii) The set of all integers between 1 and 2.
Singleton Set
A set which has only one element is called a singleton set.
For example,
(i) A = {x : 3 < x < 5, x ∈ N }
(ii) The set of all even prime numbers.
Finite Set
A set with finite number of elements is called a finite set.
For example,
1. The set of family members.
2. The set of indoor/outdoor games you play.
3. The set of curricular subjects you learn in school.
4. A = {x : x is a factor of 36}
Infinite Set
A set which is not finite is called an infinite set.
For example,
(i) {5,10,15,...}
(ii) The set of all points on a line.
Cardinal number of a set : When a set is finite, it is very useful to know how many elements it has.
The number of elements in a set is called the Cardinal number of the set.
The cardinal number of a set A is denoted by n(A)
Equivalent Sets
Two finite sets A and B are said to be equivalent if they contain the same number of elements.'
It is written as A ≈ B.
If A and B are equivalent sets, then n(A) = n(B)
For example,
Consider A = { ball, bat} and
B = {history, geography}.
Here A is equivalent to B because n(A) = n(B) = 2.
Equal Sets
Two sets are said to be equal if they contain exactly the same elements, otherwise they are said to
be unequal.
In other words, two sets A and B are said to be equal, if
(i) every element of A is also an element of B
(ii) every element of B is also an element of A
For example,
Consider the sets A = {1, 2, 3, 4} and B = {4, 2, 3, 1}
Since A and B contain exactly the same elements, A and B are equal sets.
Universal Set
A Universal set is a set which contains all the elements of all the sets under consideration and is usually
denoted by U.
For example,
(i) If we discuss about elements in Natural numbers, then the universal set U is the set of all Natural
numbers. U={x : x ∈ N }.
(ii) If A={earth, mars, jupiter}, then the universal set U is the planets of solar system.
Subset
Let A and B be two sets. If every element of A is also an element of B, then A is called a subset of B.
For example,
(i) {1}⊆{1,2,3} (ii) {2,4}⊄{1,2,3}
Proper Subset
Let A and B be two sets. If A is a subset of B and A≠B, then A is called a proper subset of B and
we write A ⊂ B.
For example,
If A={1,2,5} and B={1,2,3,4,5} then A is a proper subset of B ie. A ⊂ B.
Disjoint Sets
Two sets A and B are said to be disjoint if they do not have common elements.
In other words, if A∩B=∅, then A and B are said to be disjoint sets.
Power Set
The set of all subsets of a set A is called the power set of ‘A’. It is denoted by P(A).
For example,
(i) If A={2, 3}, then find the power set of A.
The subsets of A are ∅ , {2},{3},{2,3}.
The power set of A, P(A) = {∅ ,{2},{3},{2,3}}
(ii) If A = {∅ , {∅}}, then the power set of A is { ∅ , {∅ , {∅}}, {∅} , {{∅}} }.
