De Morgan’s First Law
De Morgan’s First Law
De Morgan’s First Law is a fundamental rule in set theory that simplifies the way we handle complements of unions and intersections. It connects the operations of union, intersection, and complement in a meaningful and logical manner.
This law is essential in solving problems involving set operations, Boolean algebra, and logic gates.
Table of Contents
- What is De Morgan’s First Law?
- De Morgan’s First Law Formula
- Venn Diagram Explanation
- Solved Examples
- Practice Questions
- Conclusion
- FAQs on De Morgan’s First Law
What is De Morgan’s First Law?
De Morgan’s First Law states:
The complement of the union of two sets A and B is equal to the intersection of the complements of A and B.
In symbolic form:
(A ∪ B)’ = A’ ∩ B’
Where:
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A’ is the complement of set A
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B’ is the complement of set B
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(A ∪ B)’ represents elements not in A or B
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A’ ∩ B’ represents elements not in A and not in B
De Morgan’s First Law Formula
De Morgan’s First Law:
(A∪B)′=A′∩B′(A ∪ B)' = A' ∩ B'(A∪B)′=A′∩B′
This means that the elements outside the union of two sets are the same as the elements common to the complements of both sets.
Venn Diagram Explanation
Visualizing the law through a Venn diagram helps solidify the concept.
Venn Diagram showing:
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A ∪ B
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Complement of A ∪ B
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A’ and B’ separately
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Their intersection A’ ∩ B’
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Show both sides of equation (A ∪ B)’ = A’ ∩ B’ resulting in the same shaded area
By comparing the two shaded areas of (A ∪ B)’ and A’ ∩ B’, we can confirm that both represent the same set.
Solved Examples of De Morgan’s First Law
Example 1:
Let U = {1, 2, 3, 4, 5, 6, 7, 8},
A = {3, 4, 5}, B = {4, 5, 6}
Prove that (A ∪ B)’ = A’ ∩ B’
Solution:
A ∪ B = {3, 4, 5, 6}
(A ∪ B)’ = U – (A ∪ B) = {1, 2, 7, 8}
A’ = {1, 2, 6, 7, 8}
B’ = {1, 2, 3, 7, 8}
A’ ∩ B’ = {1, 2, 7, 8}
∴ (A ∪ B)’ = A’ ∩ B’
Example 2:
Let U = {a, b, c, d, e, f, g, h},
P = {a, c, d}, Q = {a, b, f, g}
Prove that (P ∪ Q)’ = P’ ∩ Q’
Solution:
P ∪ Q = {a, b, c, d, f, g}
(P ∪ Q)’ = {e, h}
P’ = {b, e, f, g, h}
Q’ = {c, d, e, h}
P’ ∩ Q’ = {e, h}
∴ (P ∪ Q)’ = P’ ∩ Q’
Practice Questions
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Prove De Morgan’s First Law for:
U = {11, 12, 13, 14, 15, 16, 17}
A = {11, 12, 13, 15}
B = {13, 15, 16, 17} -
Show that (A ∪ B)’ = A’ ∩ B’ for:
U = {p, q, r, s, t, u, v, w}
A = {p, r, s, t}, B = {t, v, w}
Conclusion
De Morgan’s First Law is a powerful and essential tool in understanding the relationship between union, intersection, and complement in set theory. It simplifies complex expressions and enables students to break down and solve problems more logically and efficiently. By expressing the complement of a union as the intersection of complements, this law offers a valuable shortcut in set-based reasoning and Boolean algebra.
Whether you are working with Venn diagrams, simplifying logic circuits, or preparing for exams, mastering De Morgan’s Laws enhances your ability to think critically and solve problems with confidence.
Related Links :
Set Theory Symbols : Unlock the power of mathematical language by learning the essential set theory symbols
Subsets : Learn Subsets in Set Theory.Understand subsets with simple rules and examples.
Frequently Asked Questions on De Morgan’s First Law
Q1: What is De Morgan’s First Law?
It states that the complement of the union of two sets is equal to the intersection of their complements.
(A ∪ B)’ = A’ ∩ B’
Q2: How does De Morgan’s First Law apply in set theory?
It allows you to simplify the complement of a union operation into an intersection of complements, which can be easier to compute.
Q3: Is there a second law of De Morgan?
Yes. De Morgan’s Second Law states:
(A ∩ B)’ = A’ ∪ B’
Q4: How is De Morgan’s Law used in Boolean algebra?
In logic circuits and Boolean expressions:
¬(A ∨ B) = ¬A ∧ ¬B
Q5: Why is De Morgan’s First Law important?
It simplifies complex set operations and helps in mathematical proofs, digital logic design, and computer science.
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