There are two types of sets: finite sets and infinite sets. Things that we can count are known as finite, while things that we can not count are known as infinite.

**Finite Sets**

A set that has a number of countable elements is known as a finite set. We can say it is a countable set too.

For example, set D has days of the week, i.e. { Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday }

So n (D) = 7 countable elements, and hence it is a finite set.

Set M has months of a year, i.e. { January, February, March, April, May, June, July, August, September, October, November, December }

So n (M) = 12 countable elements, and hence it is a finite set.

**Cardinality of Finite Sets**

Total number of countable elements in a finite set is known as the cardinality of a set. If A is a set and a number of elements in a set A, then cardinality will be n (A) = a.

For example, in the above examples, the cardinality of set D is 7 and the cardinality of set M is 12.

**Properties of Finite Set**

**1) Union of Two Finite Sets:**

If we combine two sets then it is known as the union of two sets.

For example,

P = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

Q = { 1, 2, 3, 5, 7, 9, 11 }

Then union will be P plus U plus Q = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 }

**2) The Subset of a Finite Set:**

If all the elements are present in another set, then it is called a subset of a finite set.

For example,

P = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }

Q = { 1, 3, 5, 7, 9 }

Here Q is a subset of P because all its elements are present in P. It is written as Q C P.

**3) Power Set of a Finite Set:**

If set P has 3 numbers of elements, then the Power Set of Set P = 23 = 8. The power set of a finite set will be finite too.

For example,

S = { 1, 2, 3, 4 }

Power set of set S = 24 = 16

Non – Empty Finite Set

Sometimes a set has too many countable elements. In such a situation, only a number of elements is given, or their starting and ending will be given.

**For example, **

Set P = { number of people living in Gujarat state }

Here the total number is countable, but it is too difficult to count them. Such sets are known as a non-empty finite set.

**Empty Set**

If a set has zero elements or no elements at all, it is known as the empty set. It is represented by { } or ᶲ ( read as phi ). As we can count zero, the empty set is finite. The union of two empty seats will be empty too. The cardinality of an empty set will be zero.

For example, if P is a set of uncountable numbers then P = { }

Cardinality n(P) = 0

**Infinite Sets**

The set which is not finite is known as an infinite set. Suppose a set has uncountable elements then it is known as the Infinite Set. It is also known as an uncountable set. It is not possible to represent all elements of infinite sets. For example, set of points on the perimeter of the circle, set of natural numbers, set of a number of rays passing through a single given point etc.

**Cardinality of Infinite Sets**

As the number of elements in an infinite set is unlimited, the cardinality of the infinite set will be infinite too. For example, if A is a set of all-natural numbers

Cardinality n(A) = ∞

**Properties of Infinite Set**

1) The superset of any infinite set will be infinite too.

2) The union of any two infinite sets will be infinite too.

3) The power set of an infinite set will be infinite too.

**How to decide if a set is finite or infinite?**

If the set has a starting point and ending point, we can count the total number of elements, so it is a finite set. If a set has no start point or endpoint or start or endpoint, then it is impossible to count several elements in a set, so it is an infinite set.

For example,

If A = { 5, 10, 15, 20, 25 }, it has both start and endpoint, plus we can count a number of elements of set A too. So A is a finite set.

If B = { 5, 10, 15, 20, 25, 30, … }, it has a start point but does not have an endpoint, and we can not count a total number of elements in set B., so B is an infinite set.

If C = { … , -2, -1, 0, 1 ,2, … } it has neither start point nor end point and we can not count the number of elements in set C. so C is an infinite set.

**Conclusion**

In short, we can say that elements of finite sets will be countable or zero while elements of infinite sets will be uncountable. The finite set will have start and end while the infinite set might not have start, end or may not have any of them.

The cardinality of the finite set is n(A) = n while the cardinality of the infinite set is n(A) = ∞. The union of two finite sets will be finite, while two infinite sets will be infinite. The power set of the finite sets will be finite, while the power set of the infinite sets will be infinite.

It is easy and convenient to represent finite sets, while representing infinite sets is very difficult and represented with three dots to show continuity because of unlimited elements in a set.