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.

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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 =
**2**(including set A and φ as subsets along with each element individually or in the combination of each other.)^{n}

### 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.

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