Now that we know what a set is, we will want to see if there is any way to classify them. In other words, can we make sets of sets? Turns out there are different types of sets that we can define and classify mathematically. Here we will see the most important types of these sets. Let us know a bit more!

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**Types of Sets**

Since, a Set is a well – defined collection of objects; depending on the objects and their characteristics, there are many types of Sets which are explained with suitable examples, as follows: –

### Empty or Null or Void Set

Any Set that does not contain any element is called the empty or null or void set. The symbol used to represent an empty set is – **{} **or **φ**. Examples:

- Let A = {x : 9 < x < 10, x is a natural number} will be a null set because there is NO natural number between numbers 9 and 10. Therefore, A = {} or φ
- Let W = {d: d > 8, d is the number of days in a week} will also be a void set because there are only 7 days in a week.

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### Finite and Infinite Sets

Any set which is empty or contains a definite and countable number of elements is called a finite set. Sets defined otherwise, for uncountable or indefinite numbers of elements are referred to as infinite sets. Examples:

- A = {a, e, i, o, u} is a finite set because it represents the vowel letters in the English alphabetical series.
- B = {x : x is a number appearing on a dice roll} is also a finite set because it contains – {1, 2, 3, 4, 5, 6} elements.
- C = {p: p is a prime number} is an infinite set.
- D = {k: k is a real number} is also an infinite set.

### Equal and Unequal Sets

Two sets X and Y are said to be equal if they have exactly the same elements (irrespective of the order of appearance in the set). Equal sets are represented as X = Y. Otherwise, the sets are referred to as unequal sets, which are represented as X ≠ Y. Examples:

- If X = {a, e, i, o, u} and H = {o, u, i, a, e} then both of these sets are equal.
- If C = {1, 3, 5, 7} and D = {1, 3, 5, 9} then both of these sets are unequal.
- If A = {b, o, y} and B = {b, o, b, y, y} then also A = B because both contain same elements.

### Equivalent Sets

Equivalent sets are those which have an equal number of elements irrespective of what the elements are. Examples:

- A = {1, 2, 3, 4, 5} and B = {x : x is a vowel letter} are equivalent sets because both these sets have 5 elements each.
- S = {1
^{2}, 2^{2}, 3^{2}, 4^{2}, …} and T = {y : y^{2}ϵ Natural number} are also equal sets.

### Singleton Set

These are those sets that have only a single element. Examples:

- E = {x : x ϵ N and x
^{3}= 27} is a singleton set with a single element {3} - W = {v: v is a vowel letter and v is the first alphabet of English} is also a singleton set with just one element {a}.

### Universal Set

A universal set contains ALL the elements of a problem under consideration. It is *generally* represented by the letter U. Example:

- The set of Real Numbers is a universal set for ALL natural, whole, odd, even, rational and irrational numbers.

### Power Set

The collection of ALL the subsets of a given set is called a power set of that set under consideration. Example:

- A = {a, b} then Power set – P (A) = φ, {a}, {b} and {a, b}. If n (A) = m then generally, n [P (A)] = 2
^{m}

Thus, these are the different types of sets.

## Solved Examples for You

**Question 1: If A = {x: x is an even natural number} and B = {y: y is the outcome of a dice roll}, determine the nature of the two sets.**

**Answer :** A = {2, 4, 6, 8, 10, 12, 14, …} And B = {1, 2, 3, 4, 5, 6}. So, set A is **an infinite set** while set B is **a finite set.**

**Question 2: If X = {1, 2, 3, 4, 5}, Y = {a, e, i, o, u} and Z = {u, o, a, i, e}; determine the nature of sets.**

**Answer :** Since the pairs of sets X – Y, Y – Z as well as Z – X have the same number of elements, i.e. 5 they are **EQUIVALENT sets. **And sets Y and Z are also **EQUAL sets** because apart from having the number of elements the same, they also have the same elements, i.e. the alphabets of English vowel letters.

**Question 3: What is the classification of sets in mathematics?**

**Answer: **There are various kinds of sets like – finite and infinite sets, equal and equivalent sets, a null set. Further, there are a subset and proper subset, power set, universal set in addition to the disjoint sets with the help of examples.

**Question 4: What are the properties of sets?**

**Answer:** The fundamental properties are that a set can consist of elements and that two sets are equal, if and only if every element of each set is an element of the other; this property is referred to as the extensionality of sets.

**Question 5: What are Finite and Infinite Sets?**

**Answer:** Any set that is empty or consists of a definite and countable number of elements is referred to as a finite set. Sets explained otherwise, for uncountable or indefinite numbers of elements are called infinite sets.

**Question 6: What is a universal set?**

**Answer:** A universal set consists of all the elements of a problem under consideration. We generally represent it by the letter U. For instance, the set of Real Numbers is a universal set for all-natural, whole, odd, even, rational in addition to irrational numbers.