## Definition of Infinite Sets

The meaning of the word “**Infinite**” is “**Limitless** or **Endless**” and it may be presumed that **Infinite Sets** may have something to do with infinity.

Therefore, An **Infinite Set** may be defined as a set having an indefinite or infinite or endless number of elements. In other words, the sets that have an infinite number of elements are called Infinite sets. This means that the elements in an **Infinite Set** never end and keeps on increasing or decreasing depending upon the direction we move.

## Infinite Sets Symbol

To write the elements of an **Infinite Set**, an ellipsis symboled as (…….) may be placed at the end of the list, beginning of the list, or at both ends, to indicate that the list continues forever.

## Examples of Infinite sets

Set of Natural Numbers, N = \{1, 2, 3, 4, 5, 6, ……\}

Set of Whole Numbers, W = \{0, 1, 2, 3, 4, 5, 6, ……\}

Set of Integers, Z = \{……, -3, -2, -1, 0, 1, 2, 3, ……\}

In the first two examples, as shown in the above figure, the set of **Natural numbers** and **Whole numbers** has an infinite number of elements on the positive side or right-hand side of the **Numer Line**. These elements keep on increasing and never finish. Therefore, the set of **Natural Number** and **Whole Number** are the example of **Infinite Set**. To represent this, an **ellipsis** symboled as (……) is placed at the end of the list indicating that the list goes on up to infinity.

Similarly, in the third example, the set of **Integers** also has an infinite number of elements. Therefore, the Set of Integers is also an **Infinite Set**. However, in this case, the elements are both positive and negative and they keep on increasing or decreasing if we move to the right-hand side or left-hand side respectively. Hence, the **ellipsis** is placed on both sides of the list.

## Cardinality of an Infinite set

We are aware that the **Cardinality of a Set** is the number of elements present in the set. Since an **Infinite Set** has an infinite number of elements, therefore, the **Cardinality** of an **Infinite Set** is infinity or not defined. If, N represents the set of **Natural numbers** then,

n(N)=\infty

Similarly,

n(W)=\infty

n(Z)=\infty etc…

## Properties of Infinite Sets

The properties of an **Infinite Set** are as follows:

- The Union of Infinite set with any set is an Infinite Set.
- The Union of two infinite sets is also an Infinite set.
- The Power set of an Infinite set is always an Infinite set.
- The Superset of an infinite set is an Infinite set.

## Countably Infinite Set

The **Countably Infinite Set** may be defined as the set whose elements can be put into one-to-one correspondence with the set of Natural Numbers.

In other words, the **Countably Infinite Sets** are the sets whose elements can be counted even if it will take forever to count all the elements and we get a certain number of elements over a certain period of time.

### Example of Countably Infinite Set

Some examples of Countably Infinite Sets are listed below.

- The set of Integers.
- The set of odd Integers.
- The set of even Integers.
- The set of Prime Numbers.
- The set of Rational Numbers etc.

## Uncountably Infinite Set or Uncountable Set

The **Uncountably Infinite Set** or **Uncountable Set** may be defined as the set that can not be put into one-to-one correspondence with the Natural Numbers.

The **Uncountably Infinite Set** or **Uncountable Set** has so many elements that the set is larger than the set of Natural Numbers.

Also, a set is called **Uncountably Infinite Set** or **Uncountable Set** if the cardinal Number of the set is larger than the set of Natural Numbers.

### Example of Uncountably Infinite Set or Uncountable Set

Some Examples of **Uncountably Infinite Sets **or **Uncountable Sets** are,

- The set of Real Numbers.
- The set of Irrational Numbers.
- The set of Computable Numbers.

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