**Set Theory: **Are you getting confused in the set theory concept? Get 100% clarification of set theory, Subset, Union, Intersection and Venn Diagram with PDF.

## Set Theory

### Introduction to Set Theory

The word ‘

**Set is the definite collection of well-defined objects is called set. These well-defined objects are called Elements or members of a set.**

Every object of a set is known as an element of that set.

**Example:** A=

Where b, h, a, r, t, i is called elements of that set

### Important Notations in set Theory

- { a,b}: Used at the starting and ending of a set.
- ∈: Belongs to
- ∉: Not Belongs to
- ∅: Fi (Empty set/ Nothing/ Null set/ { })
- : -such that
- ⊂: Subset
- ⊃: Super Set
- ⊆: Subset Equals to
- ⊇: Superset Equals to
- ∪: Union
- ∩: Intersection
- Δ: Symmetric Difference
- ‘ : Complement
- ⇔: Equivalent to

### Representation of sets

#### Tabular or Roaster Method

In this method, we put all the elements within curly braces

**Example:**

- B= { mango, banana, apple}
- A= {1, 2, 3, 4, 5, 6, 7}

#### Set Builder Method

In this method, we list the property or properties, satisfied by the elements of sets.

**Example: **

- B= {x: x ∈ set of Fruits}
- A= {x: x∈ N and x ≤ 6}= {x: x is a natural number less than or equals to six}

Related Post: Trigonometric Formulas

## Types of set Theory

In the concept of set theory, we have 12 types of Set. They are as follows:

- Empty Set
- Singleton Set
- Finite Set
- Cardinal number of finite Set
- Infinite Set
- Equivalent Set
- Equal Set
- Subset
- Proper Subset
- Universal Set
- Power Set
- Complement Set

Let us discuss one by one

### 1) Empty Set/ Null Set/ Fi Set

A set is said to be null set or empty set or fi set, it its has no element. It is denoted by Ø or { }

**Example: **A= {x: x ∈ N and f(2x²+1)=0}= Ø

### 2) Singleton Set

A set is said to be Singleton set if it has only one element inside it.

**Example: **

- A= {x: x ∈ N and 3x – 9 = 0}
- B= { mango}

### 3) Finite Set

A set is said to be finite if it has a finite number of elements.

**Example:**

- A= {x: x∈ N and x< 103}
- B= { a, b. c, d, e,f, ….p, q,r}

### 4) Cardinal number of finite set

Cardinal Number of a finite set can be calculated by counting the number of elements inside the set.

**Example: **A= { 20, 30, 40, 50, 60}, then n(A)= 5

### 5) Infinite Set

A set is said to be infinite set if the number of elements are infinite or un-countable.

**Example: **A= {1, 2, 3, 4, 5, 6, 7, ……}

### 6) Equivalent Sets

Two sets are said to be equivalent if the cardinal number of two sets are same.

**Example: **Let A= {2, 6, 5, 9} and B= {p, a, u, j}

n(A)= 4 and n(B)= 4 So, A and B are equivalent sets

### 7) Equal Set

If two sets have the same elements, then the two sets will be equal sets. Here mandatory is not that they are in the same order.

**Example: **A= {r, a, j, p} and B= {p, a, j, r}

Here A and B have same elements r, a, j, p but they are in different order.

Still A=B

### 8) Subset

Let A, B be sets. We say that A is a subset of B, or B is super subset of A if every element of A is an element of B and can be written as A⊂B as subset and B ⊃A super subset.

i.e x∈ A ⇒x∈ B

**Example:** A= {1, 5, 7}, B= {1, 3, 5, 7, 9}

Here the elements of A i.e, 1, 5, 7 are also an element of set B. Then A ⊂B or B ⊃ A

If the element of A is not a subset of B then we can say that A ⊄B.

**Note: **

- Every set is a subset of itself.
- Empty set Ø is a subset of every set.

### 9) Proper Subset

If every element of A is an element of B but there is at least one element in B which is not in A, then A is a proper subset of B.

**Example: **A= {a, b} and B= { b, c, a, d}

A⊂B

### 10) Universal Set

A super set of each set is known as universal set. It is denoted by U.

### 11) Power Set

Let A be a set, then the collection of all subsets of A is known as Power set of A. It is denoted by P(A).

**Example: ** A= { a, b, c, d}, then

P(A)= { Ø, {a}, {b}, {c}, {d}, {a, b}, {b, c}, {c, d}, {d, a}, {a, b, c, d}}

### 12) Complement Set

The complement of a set A is the set of elements which belong to the universal set and do not belong to set A. It is denoted by A’.

**Example: **A= {p, q, r, s, t, u, v} and B= {p, q, r, s, t}

So A’= { u, v}

## Operations on Set Theory

### Union of Sets

The Union of two sets A and B is the set of elements which belong to A or B or to both A and B. It is denoted by A∪B.

**Example: **A= {1, 5, 6} and B= {2, 5, 7. 8}

So, A ∪ B= {1, 2, 5, 7, 8}

#### Venn Diagram of Union

### Intersection of Sets

Intersection of two sets A and B is the set of elements common to both A and B. It is denoted by A ∩ B.

**Example: **Let A= {m, n, o, p} and B= {p, q, r, s} then A ∩ B= {p}

#### Venn Diagram of Intersection

### Difference of Sets

Let A and B are two non-empty sets, then their difference is A-B or B-A.

A-B is a set of elements of A, which do not belong to B and B-A is a set of elements of B, which is not belonging to A.

**Example: **A= { 8, 9, 10, 11} and B= { 9, 11, 13, 12}

then A-B= { 8, 10,} and B-A= { 12, 13}

#### Venn Diagram for Difference of Set

### Symmetric Difference of sets

Let A and B are two sets, then the union of A-B and B-A is known as symmetric difference of sets A and B. It is denoted by A Δ B= (A-B) ∪ (B-A)