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-

A⊆B 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,

X⊆Y and Y⊆X ⇔ 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,

A⊂B and B⊂A ⇒ A⊂C

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:

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.

More Readings

Subsets