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A union B Complement

A union B complement is a formula in set theory that is equal to the intersection of the complements of the sets A and B. Mathematically, the formula for A union B Complement is given by, (A U B)' = A' ∩ B' or (A U B)^c = A^c ∩ B^c, where ' or ^c denote the complement of a set. This formula of A union B complement is named after the mathematician De-Morgan as one of De-Morgan's Laws of Union of Sets. The statement of this law is given as 'The complement of the union of two sets is equal to the intersection of the complements of the two sets.'

Further in this article, we will explore the A union B complement formula in detail with the help of its Venn diagram, and formula. We will also consider a few examples of sets and determine A union B Complement for a better understanding of the concept.

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

What is A union B Complement?

A union B complement is an important De-Morgan's Law of Union and is equal to the intersection of the complement of the set A and the complement of the set B. We have two laws of De-Morgan, namely A union B complement and A intersection B complement. In this article, we will mainly focus on the A union B complement formula. The formal statement of this law is given as: The complement of the union of two sets A and B is equal to the intersection of the complements of the two sets A and B.

A union B Complement Venn Diagram

Now that we know that A union B complement is equal to the intersection of A' and B', let us now understand its concept visually with the help of A union B complement Venn diagram. The diagram given below shows the universal set U with two sets A and B in it. Now, the shaded portion in blue indicates the region covered by the A union B complement. The shaded portion in blue consists of elements of the universal set U which does not include any element of the sets A and B. Hence, we can say that the blue region highlights the elements of the intersection of the complements of the two sets A and B.

A union B Complement Formula

Next, we will determine the formula for A union B complement. As we know that A union B Complement consists of those elements of the universal set U which are not in A U B, therefore, the required formula can be written in any of the following forms:

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

, where ' or ^c denote the complement of a set.

Proof of A union B Complement

Now we know that A union B complement is equal to the intersection of the complement of the set A and the complement of the set B, that is, (A U B)' = A' ∩ B'. Therefore, we will now prove this formula of A union B Complement by showing (A U B)' and A' ∩ B' as subsets of each other. For this, we will consider an arbitrary element in each of these sets.

Proof: Assume a to be an arbitrary element that belongs to (A U B)'

⇒ a ∈ (A U B)'

⇒ a ∉ (A U B) [Because an element belonging to the complement of a set cannot belong to the set]

⇒ a ∉ A and a ∉ B

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

⇒ a ∈ A' ∩ B'

⇒ (A U B)'' ⊆ A' ∩ B' --- (1)

Next, let us assume b to be an arbitrary element in A' ∩ B'

⇒ b ∈ A' ∩ B'

⇒ b ∈ A' and b ∈ B'

⇒ b ∉ A and b ∉ B [Because an element belonging to the complement of a set cannot belong to the set]

⇒ b ∉ A U B

⇒ b ∈ (A U B)'

⇒ A' ∩ B' ⊆ (A U B)' --- (2)

From (1) and (2), we get (A U B)' = A' ∩ B'. Hence, we can say that A Union B Complement is equal to the intersection of the complements of the two sets A and B.

Important Notes on A Union B Complement

A union B complement is named after the mathematician De-Morgan as one of De-Morgan's Laws of Union of Sets.
A Union B Complement is equal to the intersection of the complements of the two sets A and B.
The formula for A union B Complement is given by, (A U B)' = A' ∩ B' or (A U B)^c = A^c ∩ B^c

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A union B Complement Examples

Example 1: Verify the A union B complement formula (A ∪ B)' = A' ∩ B' for the sets A = {10, 11, 12, 13, 15}, B = {10, 12, 14} and U = {10, 11, 12, 13, 14, 15, 16, 18}

Solution: We need to prove (A ∪ B)' = A' ∩ B'. For this,

A ∪ B = {10, 11, 12, 13, 14, 15}

(A ∪ B)' = U - (A ∪ B)

= {16, 18} --- (1)

A' = U - A

= {14, 16, 18}

B' = U - B

= {11, 13, 15, 16, 18}

A' ∩ B' = {16, 18} --- (2)

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

Answer: Hence, we have verified the A union B complement formula (A ∪ B)' = A' ∩ B'
Example 2: Determine the elements of A union B complement if U = {1, 2, 3, 4, 5, 6, 7}, A = {2, 4, 6}, and B = {1, 3, 5}

Solution: We have A = {2, 4, 6}, and B = {1, 3, 5}, then A U B is given by,

A U B = {1, 2, 3, 4, 5, 6}, then A union B complement is given by,

(A ∪ B)' = U - (A ∪ B)

= {1, 2, 3, 4, 5, 6, 7} - {1, 2, 3, 4, 5, 6}

= {7}

Answer: (A ∪ B)' = {7}

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A union B Complement Questions

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FAQs on A union B Complement

What is A union B Complement in Math?

A union B complement is a formula in math that is equal to the intersection of the complements of the sets A and B. Mathematically, the formula for A union B Complement is given by, (A U B)' = A' ∩ B'

What is the Formula of A union B Complement?

The formula for A union B Complement can be written in two ways:

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

, where ' or ^c denote the complement of a set.

How to Find A union B Complement?

A union B Complement can be evaluated using its formula. As we know that the complement of the union of two sets is equal to the intersection of the complements of the two sets, therefore A union B Complement is equal to the intersection of the complements of sets A and B.

Why is A union B Complement called the De-Morgan's Law of Union?

We have two main laws of De-Morgan, namely De-Morgan's Law of Union and De-Morgan's Law of Intersection. De-Morgan's Law of Union is nothing but A union B complement formula given by (A U B)' = A' ∩ B' and De-Morgan's Law of Intersection is A intersection B complement formula which is given by, (A ∩ B)' = A' U B'.

How to Do You Prove A union B Complement Formula?

We can prove the A union B Complement Formula (A U B)' = A' ∩ B' by proving both the sets on each side of the equality as subsets of each other. We can do this by considering an arbitrary element in each set and showing that it belongs to the other set.

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