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# Subsets

## Definition of Sets

To understand the concept of subsets, first we need to recall, what is sets? A set is a collection of different things having some common property, not necessarily. The objects of a set are called the elements of a set. These elements have their own identity separately but collectively they make a set.

We have a set of 9 polygons here, but we can make a set of regular polygons only also. Or we can make a set of triangles also. So, we must be able to differentiate all elements from each other.

## Definition of Subsets

A set A is said to be a subset of set B, if all the elements of set A are the element of set B also. Or we can say that a Set A is the subset of set B if x belongs to A means X belongs to B also.

Here all the elements of set A are there in set B. So, A is a subset of B. We can also say that B contains A, so B is the superset of A.

Example

A = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}

B = {Monday, Tuesday, Wednesday}

Here, all the elements of B are present in A, so B is the subset of A.

## Symbol of Subset

To represent the subset we use a symbol it means “is a subset of”.

As in the above example B⊆A i.e. B is the subset of A.

For the superset we use the symbol “  it means “ is a superset of”.

A ⊇B i.e. A is the superset of B or A contains B.

If X is not a subset of Y then we write it with Symbol it means “ is not a subset of”.

We write it as X⊈Y i.e. X is not a subset of Y.

Symbolically we can write the definiton of subset as follows-

AB if x A  x B

We can read it as “A is the subset of B if x belongs to A implies that x also belengs to B.

## Proper Subset

If X is the subset of Y but Y is not the subset of X, then X is the proper subset of Y .Or If all the elements of X are present in Y but all the elements of Y are not there in X. Then X is the proper subset of Y.And Y is the proper superset of X.

We denote it with the symbol it means “is a proper subset of”.

And it means “is a proper superset of”.

Example

Here,

A = {1, 2, 3, 4, 5, 6}

B = {4, 5, 6}

It shows that all the elements of B are the elements of A also but all the elements of A are not there in B. So B⊂A i.e. B is the proper subset of A.

## Relation between Subset , Proper Subset and Superset A = {1,9,11}

B = {1,4,8,9,11}

C = {1,4,8,9,11}

D = {2,3}

This shows that:

• A⊂B i.e. A is the proper subset of B as All the elements of A are the elements of B but all the elements of B are not the elements of A.

• B⊃A i.e. B is the proper superset of A as B contains A but A does not contain B.

• B⊆C i.e. B is the subset of C but not the proper subset of C as B = C.

• C⊇B i.e. C is the superset of B as C contains B.

• D⊈B i.e. D is not the subset of B as all the elements of B and D are different.

## Is Every Set a Subset of itself?

If X is the subset of Y, then all the elements of X are there in Y but it is not necessary that all the elements Y are also there in X . But if It happens than Y is also the subset of X. It means that X and Y are same sets i.e. equal sets.Two sets are said to be equal sets if there elements and the number of elements are exactly same.

Symbolically,

XY and YX X=Y

“⇔” it represents the two way implication, it means “if and only if”.

We read it as X is the subset of Y and Y is the subset of X if and only if X is equal to Y.

As in the above picture it shows that X and Y have same elements , so they are equal sets . It simply means that both are same sets. As X =Y, X⊆X and Y⊆Y.

So every set is a subset of itself.

## Empty Set is a Subset of every set

Empty Set is a set with no element in it .It is a set with zero cardinality i.e. the number of element is zero (0). The symbol of empty set is { } or ∅ (phi).It is also called Null or Void Set.

If all the elements of a set X are present in set Y, then X will be a subset of Y, so if all the elements of the empty set are present in any set then it will be a subset of that particular set

Example

X= {1, 3, 5, 7}

Here, Set with 0 elements in this set is 1 i.e.-∅ (empty set)

As empty set is a set which have no element in it, so it can be easily present in any set.

This shows that empty set is the subset of every set.

∅⊆X

Empty set is a subset of every set and every set is a subset of itself. So, Empty set is also a subset of itself.

∅⊆∅

But the empty set is not the proper subset of itself. The empty set is a proper subset of all sets except .

∅ ⊂X≠ ∅

We read it as; empty set is a proper subset of X which is not equal to empty set.

## Other Important point about Subset

Here in the above picture, it shows that A is the subset of B as all the elements of A are there in B and B is the subset of C as all the elements of B are there in C.

It represents that if,

AB and BA AC

We will read it as If A is the subset of B and B is the subset of C it implies that A is the subset of C also.

Example

X = {0,1,2,3,5,6}

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

Z = {2,3,4}

Here,

Z⊂Y and Y⊂X ⇒ Z⊂X

It shows that Z is the subset of Y and Y is the subset of X so Z is also the subset of X.

## Number of Subsets

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

Here set V has 5 elements. Let’s see the possible subsets of this set:

 ∅ {1} {1,2}, {1,3} {1,2,3}, {1,2,4} {1,2,3,4,5} {2} {1,4}, {1,5} {1,2,5}, {1,3,4} {3} {2,3}, {2, 4} {1,3,5}, {1,4,5} {4} {2,5}, {3, 4} {2,3,4}, {2,3,5} {5} {3,5}, {4,5} {2,4,5}, {3,4,5}

 Number of subsets Subsets with 0 element 1 Subsets with 1 element 5 Subsets with 2 elements 10 Subsets with 3 elements 10 Subsets with 4 elements 5 Subsets with 5 elements 1 Total number of subsets 32

Here we have listed all the subsets of set V as it has 5 elements only but it is not possible to list all the subsets of a set having big number of elements.So we need some formula to calculate the number of subet.

## Formula for calculating the number of Subsets

If we will try to list the number of subsets , we can see that:

 Number of Subsets Set with 1 element 2 Set with 2 elements 4 Set with 3 elements 8 Set with 4 elements 16 Set wth 5 elements 32

This shows that every time as the number of element of a set is increasing the number of subsets is getting doubled .so the formula for the number of subsets will be 2n if the number of elements of a set is n.

As in the above example we have seen that all the subsets are the proper subsets of Set V except {1, 2, 3, 4, 5} itself. So it is true for every set that the formula for the number of proper subsets will be 2n-1, for the given number of n elements.

And sometimes people use the online algebra calculator to calculate the number of subsets.

Example

Calculate the number of subsets and proper subsets of Set M={1,3,5,7}.

Solution

Here, n(M)=4

Number of subsets =2n

=24

=16

Number of proper subsets =2n-1

=24-1

=16-1

=15

## Subsets of Set of real numbers

Real numbers is a set of all possible numbers in the universe.So all the other sets of numbers are the subsets of real numbers.

Subsets