# A Intersection B Complement – Definition, Formula, Examples, FAQs

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## What Is ‘A Intersection B Complement’?

The term “A intersection B complement” refers to the set denoted by A ∩ B’. It represents the elements that are in set A but not in set B. It signifies the intersection of set A with the complement of set B.

In other words, it includes the elements that belong to A but are not shared with B. This concept is commonly used in set theory and can help analyze relationships between different sets.

A set is defined as the collection of well-defined objects.

What is A ⋂ B?

The intersection of two sets, indicated by “A ∩ B”, is the collection of elements shared by both sets A and B. In other words, it contains all of the elements found in both sets.

What is the complement of a set?

The complement of a set includes all elements that are not in set A but present in the universal set U. In simpler language, the complement of A comprises everything in the universal set that is not A.

Complement of the set A = A’ = A̅ = AC

A’ = U-A

Keeping these definitions in mind, we can easily understand the definition of “A Intersection B Complement” or A ∩ B’. It denotes the set containing all elements from the set A and the complement of the set B.

## ‘A Intersection B Complement’ Formulas

The ‘A Intersection B Complement’ formula can be defined as

A ∩ B’ = {x | x ∈ A and x ∉ B}

We can also define the formula as

A – B = A ∩ B’

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

B’ = {1, 2, 3, 9, 10}

A ∩ B’ = {x | x ∈ A and x ∉ B}

A ∩ B’ = {1, 2, 3} …(i)

Also, A – B = {1, 2, 3, 4, 5} – {4, 5, 6, 7, 8}

A – B = {1, 2, 3} …(ii)

A ∩ B’= A – B

## Proof for A ∩ B’ = A – B

A – B represents the set that is formed by removing all the elements of A ∩ B from the set A.

Let x A ∩ B’.

⇔ x ∈ A and x ∈ B’

⇔ x ∈ A and x ∉ B

⇔ x ∈ A but x ∉ B

⇔ x ∈ (A – B)

Thus, A – B = A ∩ B’

## ‘A Intersection B Complement’ Venn Diagram

Take a look at the Venn diagram of A intersection B complement.

The shaded portion represents elements that are in set A but not in B. It defines the intersection of A and complement of B.

## A Intersection B Whole Complement

We define A intersection B whole complement as (A ∩ B)’. It represents the complement of the set (A ∩ B). It is the complement of the intersection of sets A and B.

This refers to the set of all elements in the universal set that are not present in the intersection of sets A and B. In other words, it includes all the elements that do not belong to both A and B simultaneously.

By the De Morgan’s Law, we have

(A ∩ B)’ = A’ ∪ B’

The “A intersection B whole complement” is equal to the union of A complement and B complement.

## Facts about ‘A Intersection B Complement’

• A Intersection B Complement has all elements present in the universal set except for the elements in A and not in B.
• The intersection and complement operations have significant importance in multiple areas of mathematics, such as set theory, logic, etc.

## Conclusion

In this article, we learned about the set operation A ∩ B’, where we explored the elements that belong to set A but not to set B. Understanding this concept is pivotal for grasping the relationships between sets. Let’s enhance our understanding by solving examples and practicing MCQs.

## Solved Examples on ‘A Intersection B Complement’

Example 1: There are two sets, A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}. What will be the A ∩ B’?

Solution:

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

B = {4, 5, 6, 7, 8}

A ∩ B’ = A – B

A ∩ B’ = {1, 2, 3}

Example 2: In set A = {Apple, Banana, Orange, Mango} and set B = {Banana, Peach, Grapefruit}. Find A Intersection B Complement.

Solution:

A = {Apple, Banana, Orange, Mango}

B = {Banana, Peach, Grapefruit}

A ∩ B’ = A – B

A ∩ B’= {Apple, Orange, Mango}

Example 3: What will be A ∩ B’ if A = {2, 4, 6, 8, 10}, B = {3, 6, 9}, and U = {1, 2, 3, …, 10}?

Solution:

A = {2, 4, 6, 8, 10}

B = {3, 6, 9}

U = {1, 2, 3, …, 10}

B’ = U – B = {1, 2, 4, 5, 7, 8, 10}

A ∩ B’ = {x | x ∈ A and x ∈ B’}

A ∩ B’ = {2, 4, 8, 10}

A ∩ B’ = {2, 4, 8, 10}

## Practice Problems on ‘A Intersection B Complement’

1

### Find A ∩ B' of the sets A = {1, 2, 3, 4, 5} and B = {4, 5, 6, 7, 8}.

{4, 5}
{1, 2, 3, 6, 7, 8}
{1, 2, 3}
{6, 7, 8}
CorrectIncorrect
Correct answer is: {1, 2, 3}
A ∩ B' = A - B = {1, 2, 3}
2

### A ∩ B'=

A ∩ B'={x | x ∉ A and x ∉ B}
A ∩ B'={x | x ∉ A and x ∈ B}
A ∩ B'={x | x ∈ A and x ∈ B}
A ∩ B'={x | x ∈ A and x ∉ B}
CorrectIncorrect
Correct answer is: A ∩ B'={x | x ∈ A and x ∉ B}
A ∩ B'={x | x ∈ A and x ∉ B}
3

### (A ∩ B)' =

(A ∩ B)'=A' U B'
(A ∩ B)'=A' ∩ B'
(A ∩ B)'=A ∩ B
(A ∩ B)'=A U B
CorrectIncorrect
Correct answer is: (A ∩ B)'=A' U B'
(A ∩ B)'=A' U B'

## Frequently Asked Questions about ‘A Intersection B Complement’

A universal set is a collection of all the items relevant to a specific context. It serves as a starting point for considering additional sets in that context. The universal set is a subset of all other sets under discussion. A universal set is often expressed in notation by the sign ξ (Xi) or U.

(A ∩ B)’: The complement of the intersection of sets A, and B comprises all the elements that are not shared by both sets A and B but might belong to a bigger universal set. This is also known as the complement of the intersection of A and B.

A ∩ B’: This refers to the intersection of set A with the complement of set B. It encompasses all elements that belong to set A but not set B. This is known as the intersection of A with the complement of B.

Yes, all elements in A ∩ B’ are part of set A. They’re the elements that fulfill the criteria of being in A and not in B.