Solved Examples on De Morgan’s Law
Problem 1: Given that U = {2, 3, 7, 8, 9}, A = {2, 7} and B = {2, 3, 9}. Prove De Morgan’s Second Law.
Solution:
U = {2, 3, 7, 8, 9}, A = {2, 7} and B = {2, 3, 9}
To Prove: (A ∩ B)’ = A’ ∪ B’
(A ∩ B) = {2}
(A ∩ B)’ = U – (A ∩ B) = {2, 3, 7, 8, 9} – {2}
(A ∩ B)’ = {3, 7, 8, 9}
A’ = U – A = {2, 3, 7, 8, 9} – {2, 7}
A’ = {3, 8, 9}
B’ = U – B = {2, 3, 7, 8, 9} – {2, 3, 9}
B’ = {7, 8}
A’ ∪ B’ = {3, 8, 9} ∪ {7, 8}
A’ ∪ B’ = {3, 7, 8, 9}
(A ∩ B)’ = A’ ∪ B’
Problem 2: Given that U = {1, 4, 6, 8, 9}, A = {1, 9} and B = {4, 6, 9}. Prove De Morgan’s First Law.
Solution:
U = {1, 4, 6, 8, 9}, A = {1, 9} and B = {4, 6, 9}
To Prove: (A ∪ B)’ = A’ ∩ B’
(A ∪ B) = {1, 4, 6, 9}
(A ∪ B)’ = U – (A ∪ B) = {1, 4, 6, 8, 9} – {1, 4, 6, 9}
(A ∪ B)’ = {8}
A’ = U – A = {1, 4, 6, 8, 9} – {1, 9}
A’ = {4, 6, 8}
B’ = U – B = {1, 4, 6, 8, 9} – {4, 6, 9}
B’ = {1, 8}
A’ ∩ B’ = {4, 6, 8} ∩ {1, 8}
A’ ∩ B’ = {8}
(A ∪ B)’ = A’ ∩ B’
Hence Proved
Problem 3: Simplify the Boolean Expression: Y = [(A + B).C]’
Solution:
Y = [(A + B).C]’
Applying De Morgan’s law (A . B)’ = A’ + B’
Y = (A + B)’ + C’
Applying De Morgan’s law (A + B)’ = A’. B’
Y = A’. B’ + C’
Problem 4: Simplify the Boolean Expression: X = [(A + B)’ + C]’
Solution:
X = [(A + B)’ + C]’
Applying De Morgan’s law (A + B)’ = A’. B’
X = [(A + B)’]’ . C’
X = (A + B). C’
Check this sources for more:
Topic for Interlinking | Related to |
---|---|
Boolean Algebra | De Morgan’s Law Boolean Algebra |
Set Theory | De Morgan’s Law in Set Theory |
Logical Gates | De Morgan’s Law Logic |
Discrete Mathematics | De Morgan’s Law Discrete Math |
Java Programming Examples | De Morgan’s Law Java |
De Morgan’s Law – Theorem, Proofs, Formula & Examples
De Morgan’s law is the most common law in set theory and Boolean algebra as well as set theory. In this article, we will learn about De Morgan’s law, De Morgan’s law in set theory, and De Morgan’s law in Boolean algebra along with its proofs, truth tables, and logic gate diagrams. The article also includes the solved De Morgan’s Law Example and FAQs on De Morgan’s law. Let us learn about De Morgan’s law.
Table of Content
- What is De Morgan’s Law
- De Morgan’s Law in Set Theory
- First De Morgan’s Law
- Second De Morgan’s Law
- Proof Using Algebra of Sets
- De Morgan’s Law in Boolean Algebra
- De Morgan’s Law Formula
- Solved Examples on De Morgan’s Law
- Logic Applications of De Morgan’s Law