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103 LearnersLast updated on September 26, 2025
A Intersection B Complement
Every element in the universal set, except for those that are in both A and B, is the (A∩B) complement. De Morgan’s Law states that (A ∩ B)' = A' ∪ B', meaning it includes all elements that are neither in A nor in B.
What is A ∩ B Complement?
De-Morgan’s law of the intersection of sets explains the A ∩ B complement. It can be explained by the fact that the complement of two sets’ intersections is equal to the union of their individual sets. All elements in the universal set U that are not present in both A and B are included in A ∩ B.
How to Represent A ∩ B Complement in Venn Diagram?
The Venn diagram below shows the complement of A ∩ B, or the elements that are not a part of the intersection of sets A and B. The pink area shows A ∩ B complement, which is the universal set U with all the elements of set A ∩ B removed. The blue area shows the elements of set A intersection B.
A ∩ B Complement Formula
The union of the complements of sets A and B is the same as the A ∩ B complement, as we have learned thus far in this article. Thus, the formula for the complement of A ∩ B can be expressed in any of the following ways:
- (A ∩ B)' = A' ∪ B'
- (A ∩ B)c = Ac ∪ Bc
Proof of A ∩ B Complement
We are aware that the following formula provides the complement of the intersection of two sets, A and B:
(A ∩ B)'=A' ∪ B'
We will demonstrate that each set is a subset of the others using the assumption method, also known as the method of mutual inclusion:
(A ∩ B)' ⊆ A' ∪ B' and A' ∪ B' ⊆ (A ∩ B)'.
Assume that an element x ∈ (A ∩ B)' is a part of (A ∩ B)'. This indicates that x is neither in A nor in B (i.e., x ∉ A or x ∉ B). Consequently, x ∈ (A ∩ B)', demonstrating that A' ∪ B' ⊆ (A ∩ B)'. Given that both sets are subsets of one another, we deduce that, (A ∩ B)'=A' ∪ B'
Proof: Let x be any element that is in (A ∩ B)'
⇒ x is in (A ∩ B)'
⇒ x ∉ (A ∩ B)[using the complement of a set]
⇒ x is in A' or x is in B' [using the definition of a set’s complement, x is either in A' or B']
x ∈ A' ∪ B'
(A ∩ B)' ⊆ A' ∪ B' — (1)
Next, let us say that y is an element in A' ∪ B'
⇒y ∈ A' or y ∈ B'[either y is in A' or y is in B'
[Using the complement of a set definition] ⇒ y ∉ A or y ∉ B
⇒ y ∉ A ∩ B
⇒ y ∈ (A ∩ B)'
⇒ A' ∪ B' ⊆ (A ∩ B)' — (2)
(A ∩ B)' =A' ∪ B', is obtained from (1) and (2). The complement of sets A and B is equal to the union of their complements.
Real Life Applications on A ∩ B Complement
It demonstrates how elements not shared by both sets apply in practical contexts, like excluding individuals who satisfy both requirements.
Inventory management for clearance sales
A ∩ B' helps a store identify items for clearance sales by selecting overstocked items (set A) that are not selling well (not in set B).
Academic advising for course recommendations
A university uses A ∩ B' to identify students who have completed course X (set A) but have not taken course Y (not in set B), making them eligible for the advanced course.
To prioritize new environmental policies and monitoring:
Identifying affected areas A ∩ B' assists in identifying high-pollution areas (set A) that are not yet regulated (not in set B).
Social media analytics: Influencer identification
A ∩ B' is used to spot users who talk about a product (Set A) but don’t follow the competitor (not in Set B), making them ideal influencer candidates.
Customer segmentation for targeted marketing
A ∩ B' is used to find customers who purchased product N but not product M, allowing marketers to suggest product M to them for a subsequent purchase.
Common Mistakes and How to Avoid Them in A ∩ B Complement
Students can find dealing with set complements a little confusing and make avoidable errors. Here are some common errors and ways to avoid them.
Mistake 1
Confusing B' with B
The complement of B', is frequently mispronounced by students as B.
Always keep in mind that the universal set B' contains all elements that are not in B.
Example: B' = {c, b, e} if U = {c, y, e, l, m} and B = {y, l}.
However, students might incorrectly state that B' = {y, l}, which is simply B and not its complement.
Mistake 2
Assuming A ∩ B' equals A
Without verifying which elements overlap with B, some people assume that A ∩ B' is simply A. Remember, don’t include anything that isn’t in A.
Example: B' = {1, 3, 5, …} if A = {1, 4, 3} and B = {4, 8}. Therefore, instead of {1, 4, 3} A ∩ B' = {1, 3}
Mistake 3
Ignoring the universal set (U)
Without knowing U, students might attempt to find B', which could result in an inaccurate or partial response. Keep in mind that, to find accurate B', always define the universal set (U) first.
Example: B' = {pink, blue} if U = { pink, purple, green, blue} and B = {purple, green}. Accurately determining B' is impossible without U.
Mistake 4
Mixing up A ∩ B' and (A ∩ B)'
Students think that (A ∩ B)' (everything not in both A and B) for A ∩ B' (elements in A and not in B). Use parentheses with caution. Use De Morgan’s law only when an operation has a complete complement.
Example: Let us assume, U = {3, 4, 5, 6, 7, 8}, B = {6, 7} and A = {3, 7}. Since 3 ∈ A and 3 ∉ B, A ∩ B' = {3}. (A ∩ B)' =U-{7}={3, 4, 5, 6, 8}
Mistake 5
Misusing De Morgan’s law
Students mistakenly think that De Morgan's law equals A' ∪ B' when applied to individual complements, such as A ∩ B' Always know that, only when there is a complement over the entire expression, such as in, (A ∩ B)'=A' ∪ B'Does De Morgan’s law apply?
Example: Let A = {a, b}, B = {b, c}, and U = {a, b, c, d}. B' = {a, d}. A ∩ B' = {a} However, A' ∪ B' = {c, d} ∪ {a, d} = {a, c, d} ≠ A ∩ B'
Solved Examples of A ∩ B Complement
Problem 1
In a class of 60 students, 40 play football (Set A), 19 play cricket (Set B), and 13 play both. How many students (A ∩ B') only play football?
Only football = 60 − 13 = 47
Explanation
Students who play football but not cricket receive an A ∩ B'. We deduct 13 students who play both from the total number of football players, resulting in 47 students who play football exclusively.
Problem 2
Of the 200 clients, 110 purchased product A, 60 purchased product B, and 20 purchased both. How many consumers only purchased product A?
Only product A = 110 − 20 = 90
Explanation
Because customers who bought product A but not B are, A ∩ B', we subtract those who bought both from the total number of A buyers.
Problem 3
A company has 70 employees who know Python (set A), 30 who know Java (set B), and 15 who know both. How many workers are solely proficient in Python?
Only python = 70 − 15 = 55
Explanation
Subtracting the number of employees who know both Python and Java from the total number of Python-skilled employees yields A ∩ B', which includes employees who know Python but not Java.
Problem 4
60 of the 120 students read non-fiction books (set B), 50 read fiction books (set A), 30 read both. What percentage of students only read fiction?
Only fiction = 50 − 30 = 20
Explanation
We deduct those who read both from the fiction readers because students in A ∩ B' read fiction but not non-fiction.
Problem 5
In an 80-person survey: 40 prefer tea (set A). 20 prefer coffee (set B). 10 enjoy both coffee and tea. How many people only enjoy tea?
Only tea = 40 − 10 = 30
Explanation
Those who prefer tea to coffee are represented by an A ∩ B'. After deducting the 10 people who enjoy both from the total number of tea lovers, only 30 people will enjoy tea.
FAQs On A Intersection B Complement
1.What is A ∩ B' ?
2.What is A ∩ B' formula?
3.What does De Morgan’s law say about intersection and union complements?
4.What is meant by a universal set?
5.Whole complement of A ∩ B
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