Sets Subset and Superset in Set Theory

What is a Subset?

A set refers to an organized collection of objects. In mathematics, the set is any well-defined collection of mathematical objects. Those objects can be anything from the likes, dislikes, ages of people, simple integers, to complex scientific data. It can be a result of a simple coin toss or a dice roll, or it can be an outcome of such experiments repeated hundreds/thousands of times. Roaster form and set builder are the two ways to represent a set. Elements of the set are nothing but the objects inside that particular set. In this article, you can learn about subset and superset meaning and similar concepts. We will also know what is a proper subset and improper subset.

Basics of Sets and Subsets

As you already know, a set is an organized collection of objects grouped within {}. You can further divide them into tiny sets of its own, called as subsets. In math, a set P becomes a subset of the set Q, provided that every element of set P is also an element of set Q. If ‘x’ represents any element of set P, then you can represent it symbolically as:

If x ∈ P and x ∈ Q, then P ⊂ Q (‘⊂’ denotes ‘subset of’) and it reads, P is a subset of Q. Refer fig. 1 as below.

Also, note that the converse is also true: if P ⊂ Q and x ∈ P, then x ∈ Q holds too. 

In case the P is not a subset of Q, then you can represent it as P ⊄ Q. You should also know that when P ⊂ Q, it doesn’t mean that all elements of Q are also the elements of P. However if that happens, you can represent it as, P ⊂ Q and Q ⊂ P. It also means that P = Q. 

Symbolically,  P ⊂ Q and Q ⊂ P ⟺ P = Q. Here, ‘⟺’ stands for ‘if and only if’ (iff). 

The above condition leads to fantastic insight. As P = Q, it shows that any set is the subset of itself. As you already know that null or empty set, denoted by ϕ, doesn’t contain any elements. By referring to the above discussion, you can conclude that a null set must be a subset of itself. As it doesn’t have any element, it becomes a subset of every other non-empty set. It leads us to a conclusion that each non-empty set has at least two subsets; the empty set and itself.   

What is a Superset?

The conditions that define the superset are, P ⊂ Q and P ≠ Q. When those conditions get fulfilled, you can say that Q is the superset of P. Superset gets denoted with a symbol which is a mirror image of the ‘⊂,’ that gets used to represent a subset. 

There are two properties of a superset. First, every set is a superset of a null or empty set. It means P ⊃ ϕ because ϕ has no elements at all. Secondly, now that you know each set is a subset of itself, you can also say that each set is also a superset of itself.   

So, if Q is a super of P, then you can represent it as Q ⊃ P. Below are some examples to make the concepts even more clear. 

• P = {set of all polygons} and Q = {set of regular polygons}. Here, Q ⊂ P and Q ≠ A, hence P is the superset of Q.

• A = {1, 2, 3, 4, 5, 6} and B = {m: m<4 and m ϵ N}. Here also, set B is the subset of set A and in contrast, set A is the superset of set B. 

Solved Examples

Question 1: If set P = {Mother, sister, brother, you, father} and set Q = {You}, then how is Q ⊂ P?

Answer: Here, set P denotes your family members and set Q holds a single element, which is you. The definition of subset says each element of a subset is part of the original set. From the given information, the element ‘you’ is a part of your family, which is the set P. Therefore, Q is a subset of P, represented as Q ⊂ P. 

Question 2: If P = {a: a is an even natural number} and Q = {b: b is a natural number}, then figure out the subset here. 

Answer: As per given data, P = {2, 4, 6, 8, 10, 20, . . .} and Q = {1, 2, 3, 4, 5, 6, 7, 8, 9,. . .18, 19, 20, . . .}. As you can see, set Q includes all the elements of set P. So, P is the subset of Q or P ⊂ Q.