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A Complement Union B Complement

A complement union B complement can be understood as the union of the complements of each of the two sets. The union of the complement of set A and set B is equal to the difference of the universal set(μ) and the intersection of the two sets (A n B). Further we can express A complement union B, either in roster form or using a Venn diagram.

A'UB' = (A n B)'

Here we can derive A'UB' = μ - (A n B) = (A n B)'.Let us learn more about the properties, concepts related to A complement union B complement, with examples, FAQs.

1. What Is A Complement Union B Complement?
2. Formula Of A Complement Union B Complement
3. Concepts Related To A Complement Union B Complement
4. Examples On A Complement Union B Complement
5. Practice Question
6. FAQs On A Complement Union B Complement

What Is A Complement Union B Complement?

A compliment union B complement can be computed across a sequence of steps. First, we need to find the complement of A and the complement of B. Further, the union of the complement of the two sets is to be taken. The resultant set of A complement union B complement can be observed in the below Venn Diagram.

A Compliment Union B Compliment

The union of the complement of two sets can also be presented in roster form. Let us understand this by taking the universal set μ, set A, and set B.

μ = {1, 2, 3, 4, 5, 6,7}, A = {1, 3, 5, 6}}, B = {2, 4, 6}

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

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

A' U B' = {2, 4, 7} U {1, 3, 5, 7} = {1, 2, 3, 4, 5, 7}

Thus the above example has given a clear understanding of how to find the union of the complement of the two sets.

Formula Of A Complement Union B Complement

The concept of A compliment union B complement can be expressed as a formula, which is equal to the difference of the universal set and the set A intersection B. Also, the union of the complement of the two sets can be represented as the complement of the intersection of the two sets. These two formulas can be represented as follows.

  • A' U B' = μ - (An B)
  • A' U B' = (A n B)'

Formula Of A Compliment Union B Compliment

Before trying to understand more about A complement union B complement, let us try to know more about some of the important concepts relating to set operations.

Complement of a Set

The complement of a set A is the difference of a universal set μ and set A. The complement of a set is the elements remaining after removing the elements of the given set from the universal set. The complement of a set is written as Ac or simply as A'. Here the formula of A compliment is A' = μ - A.

Union of Sets

The union of sets is a new set formed by taking all the elements of the given sets. The union of two sets is written using the symbol 'U'. The union of two sets set A and set B is written as A U B, which is formed by taking the elements of both the sets, and the common elements of the set are written only once.

Universal Set

The universal set is a set that includes all the elements of all the sets. The universal set is represented by the symbol μ. All the sets can be considered as subsets of the universal set, and the universal set is the superset of every set.

Intersection of Sets

The intersection of two sets takes the common elements of the two sets and forms a new set. The intersection of two sets is written using the symbol 'n', and is a subset of the two given sets. The intersection of two sets set A and set B is written as A n B.

Difference of Sets

The difference between the two sets is equal to the elements remaining in the first set, after removing the common elements of the two sets. The difference between the two sets set A and set B is written as A - B and is equal to the elements remaining in set A after removing the common elements of sets A and B. Also, we can write it as A - B = A - (A n B).

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Examples on A Complement Union B Complement

  1. Example 1: Find the set A complement union B complement, given that A = {2, 4, 5, 6, 7,}, B = {3, 4, 5, 8, 9}, and μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

    Solution:

    The given sets are as follows.

    A = {2, 4, 5, 6, 7,}

    B = {3, 4, 5, 8, 9}

    μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

    The aim is to find A complement union B complement = A' U B'

    A' = μ - A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {2, 4, 5, 6, 7} = {1, 3, 8, 9, 10}

    B' = μ - B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} - {3, 4, 5, 8, 9} = {1, 2, 6, 7, 10}

    A' U B' = {1, 3, 8, 9, 10} U {1, 2, 6, 7, 10} = {1, 2, 3, 6, 7, 8, 9, 10}

    Therefore, the set A' U B' is equal to {1, 2, 3, 6, 7, 8, 9, 10}.

  2. Example 2: Find A compliment union B complement, and prove that it is equal to the complement of A intersection B.

    Solution:

    The aim is to prove that A complement union B complement is equal to the complement of A intersection B.

    A' U B' = (μ - A) U (μ - B)

    = μ - (A n B)

    = (A n B)'

    Therefore we have A'UB' = (A n B)'

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FAQs on A Complement Union B Complement

What Is A Complement Union B Complement?

The set A complement union B complement can be obtained by taking the complement of set A, the complement of set B, and then taking the union of the complements of the two sets. This set can also be obtained by taking the difference of the intersection of the two sets from the universal set. A' U B' = μ - (A n B)

How To Find A Complement Union B Complement?

The following steps help in finding A complement union B complement. First, we need to find the complement of each of the two sets. As a second step, we can find the intersection of the complement of the two sets.

  • Step-I: Find A' and B'.
  • Step-II: Find the union of A' and B'.

What Is The Formula Of A Complement Union B Complement?

The two important formulas of A complement union B complement are as follows.

  • A' U B' = μ - (A n B)
  • A' U B' = (A n B)'
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