Subsets and Supersets

We saw how we can compare and operate on sets in the previous sections. If we imagine a set like a dish the ingredients of which have come together like in some exotic dish, the question that comes to mind is that whether it is possible to take a scoop out of this dish or not? It is possible if we know what a subset is! Here we will describe these and other similar concepts. Let us begin!

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Subset and Superset

Subset

Since a set is a well – defined collection of objects or elements grouped together within braces {}, it can also be disintegrated into smaller sets of its own called the subsets. Mathematically, a set A is referred to as the subset of another set B, if every element of set A is also an element of set B. Examples –

• Consider a set X such that X = {Set of all the people living in your city} then another set Y = {Set of your family members} will be a subset of X because each member of your family is also a resident of the city you live in. Every element of set Y is a part of numerous elements belonging to set X, thus Y is definitely a subset of set X.
• Let set E = {Set of all even numbers} and let set N = {Set of all natural numbers} then set E is a subset of set N.

Browse more Topics under Sets

• Introduction to Sets
• Types of Sets
• Operations on Sets
• Venn Diagrams

Any set with ALL the elements being a part of another set is called the subset of the latter and is represented as:

Y ⊂ X

The symbol – “⊂” stands for ‘is a subset of’ or ‘is contained in.’

• It is interesting to note here, that each set is unconditionally a ⊂ of itself, i.e. A ⊂ A
• Also, since the empty or null or void set φ has no elements, therefore, φ is a ⊂ of every set.
• The relationship of one set being the other’s subset is often termed as ‘inclusion’ or ‘’
• The general formula for the number of subsets for a set A with n elements is = 2ⁿ (including set A and φ as subsets along with each element individually or in the combination of each other.)

Properties Related to Subsets

• Set A is a subset of set B, if and only if their intersection is equal to set A.

A ⊂ B » A ∩ B = A

• Set A is a subset of set B, if and only if their union is equal to set B.

A ⊂ B » A ∪ B = B

Superset

Supersets are those sets which are defined by the following conditions: A ⊂ B and A ≠ B. When these two conditions are fulfilled, B is called a superset of set A. Supersets are represented by the symbol which is the mirror image of the symbol used to represent a subset:

B ⊃ A          {B is the superset of A}

Examples:

• A = {Set of all polygons} and B = {Set of regular polygons}; in this B ⊂ A and B ≠ A, therefore, A is the superset of set B.
• X = {1, 2, 3, 4, 5, 6} and Y = {s: s < 4 and s ϵ N}; in this case, set Y is the subset of X and conversely, set X is the superset of set Y.

Properties Related to Supersets

• Every set is a superset of a null or void or empty set, i.e. A⊃φ since φ has no elements.
• Since every set is a subset of itself, then every set is also a superset of itself; A ⊃

Solved examples for You

Question 1: If Set A = {Father, Mother, You, Brother, Sister} and set B = {You}, how is B⊂ A?

Answer: Set A represents your family members and set B represents a single element, i.e. You; and by the definition of a subset, each element of a subset is included in the other set, and the element ‘You’ is a part of your family so B ⊂ A (read as B is a subset of A).

Question 2: If A = {x: x is an even natural number} and B = {y: y is a natural number}, determine who is a subset here.

Answer: A = {2, 4, 6, 8, 10, 12, …} and B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, …}. It is clear that ALL the elements of set A are included in set B so, set A is the subset of set B or A⊂ B.

Question 3: How many subsets will elements have?

Answer: There are 2 subsets of a set that exist with one element. There are 4 subsets of a set that exist with two elements. Also, there are 8 subsets of a set that exist with three elements.

Question 4: Give an example of a superset?

Answer: An example of a superset can be that if B is a proper superset of A, then all elements of A shall be in B but B shall have at least one element whose existence does not take place in A.

Question 5: Explain what is meant by an improper subset?

Answer: An improper subset refers to a subset that contains every element of the original set. In contrast, a proper subset contains elements of the original set but not all. Suppose there is a set {1, 2, 3, 4, 5, 6}. In this set, {1, 2, 3, 4, 5} is an improper subset, while {1, 2, 4} and {1} are the proper subset.

Question 6: Is it possible for a subset to be a proper subset?

Answer: Yes, every proper subset happens to be a subset. A set can be both a proper subset and a subset.