#
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, X, L and K etc.
On the other hand, the entries of a set are written in ‘Small alphabets’ i.e a,
j, e and n etc. 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 = {17,
20, 29}

B = {28,
29, 30, 31, 32.....}

C = {49,
50, 51........}

# 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}

B = {10,
12……}

C = {7,
9, 11……}

# 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;

A = Set
of nine birds

B = Set
of five universities in the UK

C = Set
of five cities in the UK

D = Set
of six colleges in the London

# 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 >
13}

B = {x |
x ∈ W ^ x >
9}

C = {x |
x ∈ N ^ x >
7}

Related post: Types of triangles with pictures

#
Types of Sets in Mathematics

# Types of Sets in Mathematics in tabular form

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 = {}

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,
11, 17, 29}

C = {x | x ∈ W ^ x > 19}

C = {x | x ∈ W ^ x > 19}

#
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

B = {17,
18, 19......}

#
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, 14}

B = {a,
x, z, d}

C = {2,
4, 15, 26}

D = {p,
l, k}

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 number of 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 = {17,
33, 41, 78}

B = {d,
m, p, l}

C = {33,
17, 41, 78}

D = {d,
m, p, l}

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
singleton set has only one element or object. Some essential examples of
singleton sets are given below;

A = {19}

B = {s}

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 = {20,
25, 27}

B = {17,
18........ 36, 37}

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 = {43, 44}

B = {42,
43....48, 49}

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 = {57,
58, 59}

B = {57,
58, 59}

A is an
improper subset of B.

A = {93,
94, 112, 118}

B = {93,
94, 100, 118, 121}

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 = {13, 23} is given below;

# 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 = {113,
114.......123, 124}

A = {115,
121, 123}

B = {114,
117, 120, 123}

Now, U is
called the universal set.

Thanks for your feedback ConversionConversion EmoticonEmoticon