Sets: Subset and Superset

Sets are basically an organized collection of objects. Sets can be either represented in roster form or set builder form. The objects that a set consists of are known as the elements of the set. These elements can be grouped to form subset of the original set. For e.g. if ‘a’ is an element of set A, this is represented as:

a ∈ A , where ∈ means “belongs to”

On the other hand if ‘b’ is not an element of A, we represent it as:

b ∉ A, where ∉ means “doesn’t belong to”

The word ‘subset’ can be correlated with the words like subdivision, subcontinent, etc. Because the common part i.e. ‘sub’ is a prefix whose appropriate meaning here is forming a part from a whole. That is what we exactly do to a set to get its subsets.

What is Subset

We have already discussed its literal meaning. Mathematical meaning is a bit technical but is the same thing, more or less.

Definition 1: If all the elements of set A are also the elements of the set B, then the set A is called the subset of set B. If a represents any element of set A, then the definition is represented symbolically by:

a ∈ A and a ∈ B, then A ⊂ B (where ‘⊂’ means ‘subset of’). 

The converse is also true. That is,

If A ⊂ B and a ∈ A, then a ∈ B.

On the other hand, if A is not a subset of B, it is represented by A ⊄ B.

If A ⊂ B, it is not at all implied that all the elements of B will also be the elements of set A. However, if that happens i.e. A ⊂ B and B ⊂ A, then it implies that A = B. This is represented by:

A ⊂ B and B ⊂ A  ⟺ A = B

where ⟺  represents if and only if (iff).

The above condition gives a wonderful insight. Since A = B, it means that any set is a subset of itself. We know that the null set or empty set which is denoted by ϕ doesn’t contain any elements. As per the above discussion, a null set will be a subset of itself. Since it doesn’t have any element, it is also a subset of every other non-empty set. This means any non-empty set will have at least 2 subsets: the empty set and itself.

What is Superset

If A ⊂ B and A ≠ B, this means that A is a proper subset of B. And, is known as the superset of set A. For e.g. all natural numbers are integers. If N and Z represents the set of all the natural numbers and integers respectively, then we can write that

N ⊂ Z

Here, N is a proper subset of Z and Z is called the superset of N. We have discussed about sets, subsets and supersets. Is it possible to tell the maximum number of subsets that can be formed from a given set? Yes. The only thing required is the number of elements of that set. Let us do this step by step.

Number of elements Example of such sets Subset of sets in 2nd column Number of subsets
 0  {ϕ}  {ϕ}  1 = 20
 1  {a}  {a},{ϕ} 2 = 21
 2  {a,b} {a},{b},{a,b},{ϕ} 4 = 22
 3  {a,b,c}  {a},{b},{c},{a,b},{b,c},{a,c},{a,b,c},{ϕ} 8 = 23
 n   {a,b,c,…z,a1,b1……}  {a},{b},{c},………{a,b,c,…z,a1,b1……},{ϕ} 2n

So, if A has n elements, the maximum number of subsets of A is \( 2^n\) . On the other hand, the maximum number of non-empty sets is equal to \(2^n~-~1\)  (excluding {ϕ} from the set of subsets). Let us take a final example to improve the understanding of the concept.

If A = { 1 , 2 , 3 } , B = {{ 1 , 2 , 3 } , 4 , 5 } and C = { { 1 , 2 , 3 } , 4 , 5 , 6 , 7 },  then only B ⊂ C because all the elements of set B are also the elements of C. A ⊄ B because the elements of A are not there in B. This may seem confusing but the set A itself is an element of set B, but this doesn’t imply that A ⊂ B. Similarly, A ⊄ C.