# Venn Diagram and Basis Operations in Set Theory

*I am a PhD student of mathematics. I have complete MS in math from the University of Pakistan and have been writing online since 2020.*

## Venn Diagram and Basis Operations in Set Theory

**Some Related Questions to Title of Article "Venn Diagram and Basis Operations in Set Theory****".**

- What is Venn diagram?
- Who introduced firstly the Venn diagram?
- What is the example of Venn diagram?
- What are the uses of Venn diagram?
- what are the basis operations in set theory?
- What is union of sets and give examples?
- what is intersection of sets and give examples?
- what is complement of sets and give example?

## Venn Diagram and Basis operations in Set Theory

## Venn Diagrams in Set Theory

**Introduction:**

A Venn diagram is a graphic presentation of sets where each set is represented by a plane with enclosed spaces. It is used for comparing and contrasting two or more things, events, persons, or ideas. It is frequently used to group differences and similarities in language arts and math classes. An English philosopher and logician **John Venn** best known for developing the Venn diagrams, which are used to illustrate category propositions and evaluate the validity of categorical arguments.

**Example of Venn diagram:**

A Venn diagram is a graphic presentation of sets where each set is represented by a plane with enclosed spaces. The other sets are represented by discs situated inside a rectangle, while the universal set U is represented by the rectangle's inside. There are three categories of sets defined by diagram like:

- Figure (1) shows that A ⊆ B.
- Figure (2) shows that the sets A and B are disjoint.
- Figure (3) shows that the two sets A and B are similar.

**Uses of Venn Diagram:**

There are some uses of Venn diagram some discussed here,

- A Venn diagram is used to visually arrange the information in order to understand the relationships between different sets of things, like similarities and differences.
- A Venn diagram is used to evaluate two or more options and clearly identify what they have in common against what might set them apart.
- A Venn diagram is used to resolve difficult mathematical problems.

**Basic Operations of Sets:**

**Definition**** of Operation of Sets:**

The set operations are performed on two or more sets, to obtain a combination of elements according to the operation done on them.

The fundamental operations of the set that are introduced in this section are:

- Union
- Intersection
- Complements

**Union of Sets:**

The union of two sets A and B, denoted by A ∪ B, is the set of all elements which belong to A or to B; that is,

A ∪ B = {x | x ∈ A or x ∈ B}

Figure 3 is a Venn diagram in which A ∪ B is shaded.

**Intersection of Sets:**

The intersection of two sets A and B, denoted by A ∩ B, is the set of elements which belong to both A and B; that is,

A ∩ B = {x | x ∈ A and x ∈ B}

Figure 2 is a Venn diagram in which A ∩ B is shaded.

**Results:**

- If A ∩ B = ∅ then A and B are disjoint sets.
- If S = A ∪ B, then S is called a disjoint union of A and B.

It is important to take account of the following properties of intersection and union.

**Property No.1:**

Every element x in A and B, x, belongs to both A and B, hence x is a part of both A and B. As a result,

A ∩ B is a subset of A and of B.

Defined as,

A ∩ B ⊆ A and A ∩ B ⊆ B

**Property No.2:**

An element x belongs to the union A ∪ B if x belongs to A or x belongs to B; hence every element in A belongs to A ∪ B, and every element in B belongs to A ∪ B.

Defined as,

A ⊆ A ∪ B and B ⊆ A ∪ B

**Important Results about Union and Intersection of Sets:**

- For any sets A and B, we have:

(i) A ∩ B ⊆ A ⊆ A ∪ B and

(ii) A ∩ B ⊆ B ⊆ A ∪ B.

2. The following are equivalent:

A ⊆ B, A ∩ B = A, A ∪ B = B.

**Examples of Union:**

**1) Example No.1**

Let us consider the two sets,

A = {1, 4, 6, 7} and B = {1, 9, 5, 0}

The union of these two sets is written as,

A ∪ B = {1, 4, 5, 7, 9, 0}

**2) Example no.2:**

Let us consider the three sets like,

A = {0, 8, 9, 7} B = {5, 6, 8, 2} and C = {6, 9, 0}

The union of these sets is defined as,

A ∪ B ∪ C = {0, 2, 5, 6, 7, 8, 9}

**Examples of Intersection of sets:**

**1) Example No.1:**

Let us consider the two sets,

A = {1, 4, 6, 7} and B = {1, 9, 5, 0}

The intersection of these two sets is written as,

A ∩ B = {1}

**2) Example No.2:**

Let us consider the three sets like,

A = {0, 8, 9, 7} B = {5, 6, 8, 2} and C = {6, 9, 0}

The intersection of these sets is defined as,

(A ∩ B) ∩ C

{8} ∩ {6, 9, 0}

{ ∅}

**Complement of a Set:**

The set that contains all the universal set components that are not from the present set is known as the complement of a set.

**Example:**

The complement of set A is a set of notes if set A is a set of all coins, which is a subset of a universal set that includes all coins and notes (which do not include coins)

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2022 Kinza Javaid**