Supersets

A collection of objects is known as a set. In mathematics, these objects need to be well-defined, like scientific data, integers or likes and dislikes of certain people. It could also be a set comprising the results of rolling dice or tossing a coin. 

A set can be further broken down into smaller sets known as subsets. When the elements in two sets are the same, one is a subset of the other. Supersets are sets that contain all the elements of the other set but are not equal to it since it has additional elements. 

Set, subset and superset meaning

We have already discussed the relationship between sets, subsets and supersets. Let us try to understand the concepts better with an example. We will also look at the symbols to denote elements and subsets used in problems. 

The symbol ‘∈’ denotes an element, and the symbol ‘⊂’ signifies a subset. If x is an element in set A, then we represent it as x ∈ A. If x is an element of set A and set B, then we denote it as x ∈ A and x ∈ B. When this condition is met, then A is a subset of B and is represented as A ⊂ B. 

It is also important to note the following relationship between subsets and elements before we proceed to supersets:

• If x is an element of A and A is a subset of B, then x is also an element of B. Using the symbols, x ∈ A and A ⊂ B, then x ∈ B.
• If A is not a subset of B, then it is represented as A ⊄ B.
• If A ⊂ B, it does not mean that set A and set B have the same elements. If they have the same elements, then A ⊂ B and B ⊂ A. We can conclude that set A = set B. This also means that sets are subsets of themselves!
• Two sets can be subsets of each other, if and only if they are equal. The ‘if and only if’ symbol is ⟺. 
• A null set with no elements is denoted by ϕ, and ϕ ⊂ of all non-empty sets. For example, set {A} has two subsets, set {A} and {ϕ}.

What are supersets?

Two conditions define supersets. They are:

• A is a subset of B or A ⊂ B and 
• A is not equal to B or A ≠ B

As per superset definition,, B is a superset of A when these two conditions are met. It means that some or all the elements of set A are elements of set B. Since a superset has all the elements in other sets, they are also known as universal sets.

Supersets have two properties:

• Non-empty sets are supersets of null or empty sets. 
• Since every set has two subsets as discussed, all non-empty sets are supersets of themselves.

Concept-wise, a superset is just the opposite of a subset. While subsets are symbolized by ⊂, supersets are represented by the symbol ⊃, the mirror image of the symbol for subsets. If there are two sets A and B with B being a superset, their relationship is represented as B ⊃ A. The following examples will make the concept of supersets clearer: 

• If B = {set of all polygons} and A = {set of concave polygons}, then B is a superset of A or B ⊃ A. The reason for this is concave polygons come under the main category ‘polygons’.
• If P = {1, 5, 7, 8, 9} and Q = {5, 7}, then P ⊃ Q. Here P is the superset of Q, and Q is the subset of P. All the elements of set Q are found in the set P.

Here are some more examples of supersets to make learning fun.

Question: What is the superset of complex numbers,irrational numbers, rational numbers, integers, whole numbers and natural numbers? 

Answer: If we take A to be the set of real numbers, then A contains both rational numbers (M) and irrational numbers (N) given in the question. Then A = M ∪ N, where ∪ denotes union. This means that irrational numbers, rational numbers, integers, whole numbers and natural numbers are part of real numbers. We can denote these as follows – 

A ⊃ R (rational Numbers), A ⊃ I (integers), A ⊃ W (whole numbers) and A ⊃ Q (irrational numbers), respectively. Since complex numbers (C) are neither rational nor irrational, they cannot be called real numbers, so C ⊄ A, i.e., A is not a superset of C.

Conclusion

• Supersets are sets that include at least all the elements present in other sets known as subsets. 
• If there are two sets, then the superset must contain all the elements in the subset and should not be equal to the subset.
• A set with at least one element (a non-empty set)is a superset of a null set or empty set.
• Every set has two subsets (including the empty set), and so they are supersets of themselves
• To understand supersets better, you should understand the concept of sets and subsets and the relation between these three.