#
Sets In Mathematics

In
Mathematics, a set is defined as;

“A
collection of well-defined and distinct objects is called set”.

The entries
of a set are known as its objects. If you want to get a clear idea about sets,
it is necessary that you should have enough idea about well-defined and distinct
objects. In Mathematics, well-defined and distinct objects mean those objects
which are not equal to other objects. For example, if we say that set of natural
numbers, the entries of the natural numbers’ set are 1,2,3,4 and so on. 1, 2, 3
and 4 are well-defined and distinct objects because 1 is not equal to 2 and
vice versa. On the other hand, if we say that the set of naughty students in
the class, we can’t distinguish between naughty students in the class. In other
words, the entries of this set can be equal to each other. As these entries of
the set are not well-defined and distinct, therefore, it is not a set.

#
How To Represent Sets in Mathematics?

There is
also a specific method to represent sets in mathematics. In order to represent sets
in mathematics, we have to use ‘Capital alphabets’ i.e A, B, C, D and so on. On
the other hand, the entries of a set are written in ‘Small alphabets’ i.e a, b,
c, d and so on. Anyhow, the entries of the sets are also written in the form of
numerals and symbols etc. These entries of a set are enclosed in the braces or
the curly brackets. Some essential examples of the sets are given below;

A = {1, 2,
3, 4}

N = {1, 2,
3, 4, 5…….}

W = {0, 1,
2, 3, 4, 5, 6…….}

#
Ways To Represent Sets in Mathematics

There are
three possible ways to represent sets in mathematics. These three ways are
explained below;

#
Tabular Form

The method
of representing the set in which we list all the elements of the set and these
elements of sets are separated by commas. After listing these elements, we
enclose these elements in the curly brackets. Some essential examples of sets
that are represented in the tabular form are given below;

A = {6, 7,
8, 9, 10}

E = {2, 4,
6, 8, 10……}

O = {1, 3,
5, 7, 9……}

#
Descriptive Form

The process
of representing the sets in the form of words is known as descriptive form.
Some essential examples of the descriptive form of sets are given below;

N = Set of
natural numbers

W = Set of
whole numbers

Z = Set of
integers

O = Set of
odd numbers

E = Set of
even numbers

#
Set Builder form

The
presentation of the sets in the symbolic form is known as set builder form.
Some essential examples of the representation of the sets in set builder form
are given below;

A = {x | x ∈ N ^ x > 10}

B = {x | x ∈ W ^ x > 5}

#
Types of Sets in Mathematics

There are different types of sets in mathematics. These types
of sets along with their examples are explained below;

#
Empty or Null Set

A set which has no element is known as an empty or null set. In
order to represent an empty or null set, we use the symbol {} or ∅. We can represent a null or an empty set
in the following way;

A = ∅

B = {}

#
Finite Set

A set which has the finite number of elements or the limited
number of elements is known as a finite set. Some essential examples of finite
sets are given below;

A = Set of first ten natural numbers

B = {1, 2, 3, 4, 5}

C = {x | x ∈ W ^ x > 5}

#
Infinite Set

A set which has the infinite number of elements or the unlimited
number of elements is known as an infinite set. Some essential examples of
infinite sets are given below;

N = Set of natural numbers

W = {0, 1, 2, 3, 4, 5……..}

#
Equivalent Sets

Two sets are called equivalent sets if these sets have the equal
number of elements. We can represent two equivalent sets by using the symbol ↔.
Some essential examples of equivalent sets are given below;

A = {1, 2, 3}

B = {a, b, c, d}

C = {2, 4, 5, 6}

D = {x, y, z}

A and D are equivalent sets because these sets have an equal
number of elements and we can represent these two sets as A ↔ D. On the other hand, B and C are
also equivalent sets. Its reason is that these two sets have an equal number of
elements and we can represent these two sets as B ↔ C.

#
Equal Sets

Two sets are
called equal sets if these two sets have equal elements and these elements are
also same. In order to represent two equal sets, we use the symbol ‘=’. Some
essential examples of equal sets are given below;

A = {1, 2,
3, 4, 5}

B = {a, b, c,
d}

C = {2, 3,
4, 5, 1}

D = {a, b,
c, d}

A and C are
equal sets because these sets have an equal number of elements and these
elements are also the same. We can represent these two sets as A = C. On the
other hand, B and D are also equal sets because these two sets have an equal
number of elements and these elements are also the same. We can also represent
these two sets as B = D.

#
Singleton Set

A set which
has only one element is known as singleton set. Some essential examples of
singleton sets are given below;

A = {11}

B = {a}

C = {0}

#
Subset

If A and B
are two sets and all the elements of set A are present in the set B, set A is
called the subset of set B. Example of the subset is given below;

A = {1, 2,
3}

B = {1, 2,
3, 4}

As all the
elements of A are present in the B, therefore, A is called the subset of B.

#
Proper subset

If A and B
are two sets and A is a subset of B and A ≠ B, A is also called a subset of B.
An example of the subset is given below;

A = {1, 2}

B = {1, 2,
3, 4}

Now, we can
say that A is the subset of B.

#
Improper Subset

If A and B
are two sets and A is an improper subset of B only either A = B or A contains
at least one such element which is not present in set B. Example of an improper
subset is given below;

A = {1, 2,
3, 4}

B = {1, 2,
3, 4}

A is an improper
subset of B.

A = {1, 2,
3, 4, 7}

B = {1, 2,
3, 4, 5}

A is improper
subset of B.

#
Power set

A set which
contains all the subsets of a set is known as its power set. The power set of A
= {1, 2} is given below;

P (A) = { ∅, {1}, {2}, {1,2} }

#
Universal Set

A set which is superset of all the sets under consideration is
known as a universal set. An example of universal set is given below;

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

A = {1, 2,
3, 4, 5}

B = {2, 4,
6, 8}

Now, U is
called the universal set.