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A Intersection B Complement

A Intersection B Complement is one of the important De-Morgan's Law of sets. As we know, A Intersection B consists of elements that are common in both sets A and B, and a complement of a set consists of all elements other than the set itself. Therefore, A Intersection B Complement consists of all elements of the universe except the elements in A Intersection B. In other words, we can say that A Intersection B Complement is equal to the union of the complements of the sets A and B. Mathematically, it is written as (A ∩ B)' = A' U B'.

In this article, we will understand the A Intersection B Complement formula with the help of its Venn diagram and prove its formula. We will also solve a few examples based on A Intersection B Complement for a better understanding of the concept.

1.	What is A Intersection B Complement?
2.	A Intersection B Complement Venn Diagram
3.	A Intersection B Complement Formula
4.	A Intersection B Complement Proof
5.	FAQs on A Intersection B Complement

What is A Intersection B Complement?

A Intersection B Complement is known as De-Morgan's Law of Intersection of Sets. It can be explained as the complement of the intersection of two sets is equal to the union of the complements of those two sets. In simple words, we can say that A Intersection B Complement consists of elements of the universal set U which are not the elements of the set A ∩ B. This implies the complement of the intersection of sets A and B is equal to the union of the sets A^c and B^c. Let us now see the Venn diagram for A Intersection B Complement to understand it visually.

A Intersection B Complement Venn Diagram

The Venn diagram given below shows the complement of the intersection of two sets A and B. The region in blue indicates A Intersection B Complement which consists of all the elements of the universal set U excluding the elements of the set A intersection B which are shaded in yellow. As we can see, the blue shaded area of A Intersection B Complement can also be expressed as the union of the complement of set A and complement of set B.

A Intersection B Complement Formula

As we have studied so far in this article, A Intersection B Complement is equal to the union of the complement of the set A and complement of the set B. Therefore, the formula for A Intersection B Complement can be written in any of the following forms, where ' or ^c indicate the complement of the set:

(A ∩ B)' = A' U B'
(A ∩ B)^c = A^c U B^c

A Intersection B Complement Proof

Now that we know the formula for A Intersection B Complement which is given by (A ∩ B)' = A' U B', let us now prove it by using the assumption method and showing the two sets (A ∩ B)' and A' U B' as subsets of each other. The proof of A Intersection B Complement Formula is as follows:

Proof: Let x be an arbitrary element that belongs to (A ∩ B)'

⇒ x ∈ (A ∩ B)'

⇒ x ∉ (A ∩ B) [Using complement of a set definition]

⇒ x ∉ A or x ∉ B

⇒ x ∈ A' or x ∈ B' [Using complement of a set definition]

⇒ x ∈ A' U B'

⇒ (A ∩ B)' ⊆ A' U B' --- (1)

Next, let us assume y to be an arbitrary element in A' U B'

⇒ y ∈ A' U B'

⇒ y ∈ A' or y ∈ B'

⇒ y ∉ A or y ∉ B [Using complement of a set definition]

⇒ y ∉ A ∩ B

⇒ y ∈ (A ∩ B)'

⇒ A' U B' ⊆ (A ∩ B)' --- (2)

From (1) and (2), we get (A ∩ B)' = A' U B', i.e., A Intersection B Complement is equal to the union of the complements of the sets A and B.

Important Notes on A Intersection B Complement

The formula for A Intersection B Complement is: (A ∩ B)^c = A^c U B^c
A Intersection B Complement consists of all elements of the universe except the elements in A Intersection B.

Related Topics on A Intersection B Complement

A Intersection B Complement Examples

Example 1: Consider the universal set U = {a, b, c, d, e, f, g, h, i, o, u}, sets A = {a, e, i, o, u} and B = {a, b, c, d, f, g, h, u}. Show that (A ∩ B)' = A' U B'.

Solution: Find A ∩ B

A ∩ B = {a, e, i, o, u} ∩ {a, b, c, d, f, g, h, u}

= {a, u}

(A ∩ B)' = U - (A ∩ B)

= {b, c, d, e, f, g, h, i, o} --- (1)

Now, A' = U - A

= {b, c, d, f, g, h}

B' = U - B

= {e, i, o}

A' U B' = {b, c, d, f, g, h} U {e, i, o}

= {b, c, d, e, f, g, h, i, o} --- (2)

From (1) and (2), we get (A ∩ B)' = A' U B'

Answer: Hence, we have proved the A Intersection B Complement formula for the given sets A and B.
Example 2: Consider U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A' = {2, 3, 5} and B' = {1, 2, 3, 4, 5}. Find the elements in A Intersection B Complement.

Solution: We know that (A ∩ B)' = A' U B'

Now, A' U B' = {2, 3, 5} U {1, 2, 3, 4, 5}

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

⇒ (A ∩ B)' = {1, 2, 3, 4, 5}

Answer: The elements in A Intersection B Complement are {1, 2, 3, 4, 5}.

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A Intersection B Complement Questions

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FAQs on A Intersection B Complement

What is A Intersection B Complement in Set Theory?

A Intersection B Complement is equal to the union of the complements of the sets A and B. Mathematically, it is written as (A ∩ B)' = A' U B'. It is one of the important De-Morgan's Law of sets.

How Do You Find the Complement of A Intersection B?

A Intersection B Complement can be evaluated using the formula (A ∩ B)' = A' U B' or using the Venn diagram.

What is A Intersection B Complement Formula?

The formula for A Intersection B Complement can be written in any of the following forms, where ' or ^c indicate the complement of the set:

(A ∩ B)' = A' U B'
(A ∩ B)^c = A^c U B^c

What is the Probability of A Intersection B Complement?

The probability of A Intersection B Complement is given by, P((A ∩ B)^c) = 1 - P (A ∩ B) or P[(A ∩ B)^c]= P(A^c U B^c)

What is De-Morgan's Law of Intersection of Sets?

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