Sets are** collections of unique objects or elements, and set operations are mathematical operations carried out on sets. **

Table of Content

## What is Set Operation

Set operations are **mathematical operations performed ****operation on sets, **** which are collections of distinct objects or elements. **These operations are fundamental in set theory and are used to manipulate sets, define relationships between sets, and solve problems involving collections of objects.

**There are three major types of operations performed on sets, such as:**

- Union of sets
- Intersection of sets
- Disjoint
- Set DifferenceÂ
- Complement
- Addition & SubtractionÂ

**Let us discuss these operations one by one.**

**Union**Â

**Union**

Union of sets A and B, denoted by ** A âˆª B, **is the set of distinct elements that belong to set A or set B, or both.

Â

**Above is the Venn Diagram of A U B.**

*Example*: Find the union of A = {2, 3, 4} and B = {3, 4, 5};
Solution : A âˆª B = {2, 3, 4, 5}.

**Intersection**Â

**Intersection**

The intersection of the sets A and B, denoted by** A âˆ© B,** is the set of elements that belong to both A and B i.e. set of the common elements in A and B.

Â

**Above is the Venn Diagram of A âˆ© B.**

*Example*: Find the intersection of A = {2, 3, 4} and B = {3, 4, 5}
Solution : A âˆ© B = {3, 4}.

**Disjoint**Â

**Disjoint**

Two sets are said to be disjoint if their intersection is the empty set. i.e, sets have no common elements.

Â

**Above is the Venn Diagram of A disjoint B.**

*Example*: Let A = {1, 3, 5, 7, 9} and B = { 2, 4, 6, 8}
A and B are disjoint sets since both of them have no common elements.

**Set Difference**Â

**Set Difference**

The difference between sets is denoted by** ‘A – B’**, which is the set containing elements that are in A but not in B. i.e., all elements of A except the element of B.

Â

Above is the **Venn Diagram of A-B.**

*Example*: If A = {1, 2, 3, 4, 5} and B = { 2, 4, 6, 8}, find A-B
Solution: A-B = {1, 3, 5}

### Complement

**The complement of a set A, denoted by A**^{C }**is the set of all the elements except the elements in A. Complement of the set A is U – A.**

Â

**Above is the Venn Diagram of A**^{c}

*Example*: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8}.
Find A

^{C}Solution: A

^{C}= U-A = {1, 3, 5, 7, 9, 10}

**Addition & Subtraction**Â

**Addition & Subtraction**

Addition of sets **A and B, referred to as ****Minkowski addition**** ,** is the set in whose elements are the sum of each possible pair of elements from the 2 sets (

**that is one element is from set A and the other is from set B).****It is to be observed that these operations are operable only on numeric data types. Even if operated otherwise, it would only be a symbolic representation without any significance. Further,**

**Set subtraction follows the same rule, but with the subtraction operation on the elements.**

**it can be seen easily that set addition is commutative, while subtraction is not.Â**Â

**For addition and consequently subtraction, please refer to this answer.**

[Tex]n(A\cup B) =n(A) + n(B) – n(A\cap B)Â Â [/Tex][Tex]A-B=A\cap \bar{B}Â Â [/Tex]

: Set Operations practiceCheck

## Property of set operation

**Set operations have several important properties that govern their behavior. Here are some fundamental properties of set operations:**

**Closure Property: **

**Closure Property:**

- Set operations are closed under their respective operations, meaning that performing an operation on sets results in another set.
- For example, the union, intersection, and difference of sets always produce sets as their results.

**Commutative Property:**

**Commutative Property:**

- Union: A âˆª B = B âˆª A
- Intersection: A âˆ© B = B âˆ© A
- Symmetric Difference: A Î” B = B Î” A

### Associative Property:

- Union: (A âˆª B) âˆª C = A âˆª (B âˆª C)
- Intersection: (A âˆ© B) âˆ© C = A âˆ© (B âˆ© C)

### Distributive Property:

- Union over Intersection: A âˆª (B âˆ© C) = (A âˆª B) âˆ© (A âˆª C)
- Intersection over Union: A âˆ© (B âˆª C) = (A âˆ© B) âˆª (A âˆ© C)

### Identity Property:

- Union: A âˆª âˆ… = A
- Intersection: A âˆ© U = A, where U represents the universal set
- Symmetric Difference: A Î” âˆ… = A

### Complement Property:

- Union: A âˆª A’ = U, where U is the universal set
- Intersection: A âˆ© A’ = âˆ… (the empty set)

### Absorption Property:

- Union over Intersection: A âˆª (A âˆ© B) = A
- Intersection over Union: A âˆ© (A âˆª B) = A

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## Conclusion of Set Operation

Set operations are essential tools that let us work with,** combine, and evaluate groups of items or components**. We may represent real-world settings, describe relationships between sets, and solve set-related problems by using

**A systematic foundation for comprehending and thinking about sets is provided by the qualities of set operations,**

**operations like union, intersection, complement, difference, and symmetric difference.**

**which include closure, commutative, associative, distributive, identity, complement, and absorption features.****Set Operations – FAQs**

**Set Operations – FAQs**

**What are the 4 operations of sets?**

The four fundamental operations of sets, commonly used in set theory and mathematics, are:

UnionIntersectionDifferenceComplement

### What are operations 4?

- They are
addition, subtraction, multiplication, and division.

### What is the union symbol?

“âˆª” serves as the union symbol, which symbolizes the process of joining two sets to create a new set that includes every unique element from both sets.

### What is power of a set?

The Set, including

is called the power set of that set. the power set of a set withthe empty set and the set itself,subsets if the set has n elements.n elements will have 2 n

### What is the formula for set operation?

n(PâˆªQ) = n(P) + n(Q) âˆ’ n(Pâˆ©Q).