A Complement Intersection B Complement
A compliment intersection B Compliment is obtained by taking the complement of the set A, set B, and then taking the intersection of the complement of the two sets. A complement intersection B complement is represented as A' n B', which is equal to (A U B)',
A' n B' = (A U B)'
Let us learn more about A complement intersection B complement, the formulas, and related concepts, with the help of examples, FAQs.
What Is A Complement Intersection B Complement?
A complement intersection B complement is the common elements of A competent and B complement. This required A'nB' can be easily understood from the Venn diagram and is equal to the complement of A U B. In a sequential manner A' n B' can be computed by first finding the complements of set A and set B respectively. The complement of set A is the part of the set B and the remaining part of the universal set μ, and the complement of set B is the part of the set A and the remaining part of the universal set μ.
This A complement intersection B complement can be easily observed from the below Venn diagram and it represents the area beyond A U B and is the complement of A U B or (A U B)'.
The intersection of the complement of two sets can also be presented in roster form. Let us try to understand this by taking a simple example of the universal set μ, set A, and set B.
μ = {1, 2, 3, 4, 5, 6,7, 8, 9, 10, 11, 12}, A = {1, 3, 5, 7, 9, 11}}, B = {2, 4, 6, 8, 10, 12}
A' = μ  A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}  {1, 3, 5, 7, 9, 11} = {2, 4, 6, 8, 10, 12}
B' = μ  B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}  {2, 4, 6, 8, 10, 12} = {1, 3, 5, 7, 9, 11}
A' n B' = {2, 4, 6, 8, 10, 12} n {1, 3, 5, 7, 9, 11} = { }
Thus the above example has given a clear understanding of how to find the intersection of the complement of the two sets.
Formula Of A Complement Intersection B Complement
The concept of A compliment intersection B complement can be expressed as a formula, which is equal to the difference of the universal set and the set A union B. Also, the intersection of the complement of the two sets can be represented as the complement of the union of the two sets. These two formulas can be represented as follows.
 A' n B' = μ  (A U B)
 A' n B' = (A U B)'
Concepts Related To A Complement Intersection B Complement
Before trying to understand more about A complement intersection 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 A^{c} or simply as A'. Here the formula of A compliment is A' = μ  A.
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.
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.
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 Intersection B Complement

Example 1: Find the set A complement intersection B complement, given that A = {1, 2, 4, 5, 6, 7, 8}, B = {3, 4, 5, 6, 8, 9, 10}, and μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.
Solution:
The given sets are as follows.
A = {1, 2, 4, 5, 6, 7, 8}
B = {3, 4, 5, 6, 8}
μ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
The aim is to find A complement intersection B complement = A' n B'
A' = μ  A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}  {1, 2, 4, 5, 6, 7, 8} = {3, 9, 10}
B' = μ  B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}  {3, 4, 5, 6, 8} = {1, 2, 7, 9, 10}
A' n B' = {3, 9, 10} U {1, 2, 7, 9, 10} = {9, 10}
Therefore, the set A' n B' is equal to {9, 10}.

Example 2: Find A compliment intersection B complement, and prove that it is equal to the complement of A union B.
Solution:
The aim is to prove that A complement intersection B complement is equal to the complement of A union B.
A' n B' = (μ  A) n (μ  B)
= μ  (A U B)
= (A U B)'
Therefore we have A' n B' = (A U B)'
FAQs on A Complement Intersection B Complement
What Is A Complement Intersection B Complement?
The set A complement intersection B complement can be obtained by taking the common elements of the complement of set A, and the complement of set B. This set can also be obtained after removing the union of the two sets from the universal set. A' n B' = μ  (A U B).
How To Find A Complement Intersection B Complement?
The required set A complement intersection B complement can be obtained by first finding the complement of each of the two sets. The complements of set A and set B are sets A', and B' respectively. Further, the common element of A' and B' are taken to find A' n B'. Also from observation and formula A' n B' = (A U B)'.
What Is The Formula Of A Complement Intersection B Complement?
The two important formulas of A complement intersection B complement are as follows.
 A' n B' = μ  (A U B)
 A' n B' = (A U B)'
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