# Sets and Types of Sets

A set is a collection of distinct objects(elements) which have common property. For example, cat, elephant, tiger, and rabbit are animals. When, these animals are considered collectively, it’s called set.

## Set Notation

The members(elements) of set is separated by comma and braces { } are used outside the comma separated elements.

`{cat, elephant, tiger, rabbit}`

For convenience, sets are denoted by a capital letter. For example,

`A = {cat, elephant, tiger, rabbit}`

Here, A is a set containing 4 elements.

### Important Points Regarding Sets

1. A Set is collection of distinct members.
```		Wrong: Elements are not distinct
A = {a, b, c, b, c, d}

Right: Elements are distinct
A = {a, b, c, d}
```
2. The number of elements in a set can be both finite and infinite.
```          N = {2, 4, 6}
N = {..., -2, -1, 0, 1, 2, ...}
```
3. The elements of a set can be in any order.Changing the order of elements doesn’t change anything.
```         A = {1, 2, 3}

This set can also be written as:
A = {2, 1, 3}
A = {3, 1, 2} and so on ```

## Example of Sets

Example #1: What is the set of all vowels in English alphabet?

`V  = {a, e, i, o, u}`

Example #2: What is the set of integers between 2 and 9?

`I = {3, 4, 5, 6, 7, 8}`

Example #2: What is the set of prime number?

```P = {2, 3, 5, 7, 11, ... }
```

## Types of Sets

### Empty set

A set which do not have any element is known as empty set. It is also called Null Set, Vacuous Set or Void Set. Empty set is denoted by ϕ.

`A = {} =ϕ`

### Singleton set

If a set has only one element, it’s known as singleton set.

`A = { moon }`

### Finite set

Set with finite number of elements is called finite set.

`S = {1, 2, 3}`

### Infinite set

A set with have infinite number number of elements is called infinite set.

```A= { x:x is an integer }
B = { 5, 10, 15, 20, 25, ... }```

### Equivalent sets

Two sets are said to be equivalent sets if they have same number of elements. For Example,

```A = {a, b, c, d}
B = {e, f, g, h}```

Here, A and B are equivalent sets because both sets have 4 elements.

### Equal Sets

Two sets are said to be equal sets if they both have exactly same elements.

```A = {1, 2, 3, 4}
B = {2, 4, 3, 1}
```

Here, A and B are equal sets because both set have same elements (order of elements doesn’t matter).

### Overlapping Sets

Two sets are said to be overlapping sets if they have at least one element common.

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

Here A and B are overlapping sets because elements 3 and 4 are common in both sets.

### Disjoint Sets

Two sets are said to be disjoint sets if they don’t have common element/s.

```A = {1, 2, 3, 4}
B = {5, 6}
```

Here A and B are disjoint sets because these two sets don’t have common element.

### Subset

A set P is a subset of set Q if every element of set P is also the member of set Q. Simply, if set P is contained in set Q, P is called subset of superset Q. It is denoted by P⊂Q.

```P = {1, 2, 3}
Q = {1, 2, 4, 3, 9}```

Here, all three elements 1, 2, and 3 of set P is also member of set Q. Hence, P is subset of Q.