along with various examples and questions.

## Definition of Finite set

Finite sets are the sets having a finite/countable number of members. Finite sets are also known as **countable sets** as they can be counted. The process will run out of elements to list if the elements of this set have a finite number of members.

**Examples of finite sets:**

P = { 0, 3, 6, 9, …, 99}

Q = { a : a is an integer, 1 < a < 10}

A set of all English Alphabets (because it is countable).

**Another example of a Finite set:**

A set of months in a year.

**M = **{January, February, March, April, May, June, July, August, September, October, November, December}

n (M) = 12

It is a finite set because the number of elements is countable.

### Cardinality of Finite Set

If ‘a’ represents the number of elements of set A, then the cardinality of a finite set is **n(A) = a**.

So, the Cardinality of the set A of all English Alphabets is 26, because the number of elements (alphabets) is 26.

Hence, n (A) = 26.

Similarly, for a set containing the months in a year will have a cardinality of 12.

So, this way we can list all the elements of any finite set and list them in the *curly braces* or in *Roster form.*

### Properties of Finite sets

The following finite set conditions are always finite.

- A subset of Finite set
- The union of two finite sets
- The power set of a finite set

**Few Examples:**

P = {1, 2, 3, 4}

Q = {2, 4, 6, 8}

R = {2, 3)

- Here, all the P, Q, R are the finite sets because the elements are finite and countable.
- R ⊂ P, i.e R is a Subset of P because all the elements of set R are present in P. So, the subset of a finite set is always finite.
- P U Q is { 1, 2, 3, 4, 6, 8}, so the union of two sets is also finite.

**The number of elements of a power set = 2 ^{n}.**

The number of elements of the power set of set P is 2^{4 }= 16, as the number of elements of set P is 4. So it shows that the power set of a finite set is finite.

### Non- Empty Finite set

It is a set where either the number of elements are big or only starting or ending is given. So, we denote it with the number of elements with n(A) and if n(A)is a natural number then it’s a finite set.

**Example**:

S = { a set of the number of people living in India}

It is difficult to calculate the number of people living in India but it’s somewhere a natural number. So, we can call it a non-empty finite set.

If N is a set of natural numbers less than n. So the cardinality of set N is n.

N = {1,2,3….n}

X = x_{1}, x_{2}, ……, x_{n}

Y = {x : x_{1} ϵ N, 1 ≤ i ≤ n}, where i is the integer between 1 and n.

### Can we say that an empty set is a finite set?

Let’s learn what is an empty set first.

An **empty set** is a set which has no elements in it and can be represented as { } and shows that it has no element.

P = { } Or ∅

As the finite set has a countable number of elements and the empty set has zero elements so, it is a definite number of elements.

So, with a cardinality of zero, an empty set is a finite set.

## What is Infinite set?

If a set is not finite, it is called an **infinite set** because the number of elements in that set is not countable and also we cannot represent it in Roster form. Thus, infinite sets are also known as **uncountable sets**.

So, the elements of an Infinite set are represented by 3 dots (ellipse) thus, it represents the infinity of that set.

### Examples of Infinite Sets

- A set of all whole numbers, W= {0, 1, 2, 3, 4,…}
- A set of all points on a line
- The set of all integers

### Cardinality of Infinite Sets

The cardinality of a set is n (A) = x, where x is the number of elements of a set A. The cardinality of an infinite set is n (A) = ∞ as the number of elements is unlimited in it.

### Properties of Infinite Sets

- The union of two infinite sets is infinite
- The power set of an infinite set is infinite
- The superset of an infinite set is also infinite

## Comparison of Finite and Infinite Sets:

Let’s compare the differences between Finite and Infinite set:

The sets could be equal only if their elements are the same, so a set could be equal only if it is a finite set, whereas if the elements are not comparable, the set is infinite.

Factors |
Finite sets |
Infinite sets |
---|---|---|

Number of elements | Elements are countable | The number of elements is uncountable |

Continuity | It has a start and end elements | It is endless from the start or end. Both the sides could have continuity |

Cardinality | n(A) = n, n is the number of elements in the set | n(A) = ∞ as the number of elements are uncountable |

union | Union of two finite sets is finite | Union of two infinite sets is infinite |

Power set | The power set of a finite set is also finite | The power set of an infinite set is infinite |

Roster form | Can be easily represented in roster form | As the set in infinite set can’t be represented in Roster form, so we use three dots to represent the infinity |

### How to know if a Set is Finite or Infinite?

As we know that if a set has a starting point and an ending point both, it is a finite set, but it is infinite if it has no end from any side or both sides.

Points to identify a set is whether a finite or infinite are:

- An infinite set is endless from the start or end, but both the side could have continuity unlike in Finite set where both start and end elements are there.
- If a set has the unlimited number of elements, then it is infinite and if the elements are countable then it is finite.

### Graphical Representation of Finite and Infinite Sets

Here in the above picture,

A = {1, 2, 3, 4, 5}

B = {1, 2, 6, 7, 8}

A U B = {1, 2, 3, 4, 5, 6, 7, 8}

A∩B = {1, 2}

Both A and B are finite sets as they have a limited number of elements.

n(A) = 5 and n(B) = 5

AUB and A∩B are also finite.

So, a Venn diagram can represent the finite set but it is difficult to do the same for an infinite set as the number of elements can’t be counted and bounced in a circle.

### Read More:

Universal Set | Power Set |

Set Operations: Intersection And Difference Of Two Sets | Set Formulas |

Union And Intersection Of Sets |