Question

Which one of the following options is not a property of Boolean Algebra? (Note: $+$ is OR operation, $\cdot$ is AND operation, and $'$ is NOT operation.)

• A. $a + b = b + a$
• B. $a \cdot (b + c) = (a \cdot b) + (a \cdot c)$
• C. $a + a = 2a$
• D. $a + (b \cdot c) = (a + b) \cdot (a + c)$

Hint:

Remember the Idempotent Laws: $x + x = x$ and $x \cdot x = x$. There are no coefficients or exponents in Boolean Algebra.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Boolean Algebra operates on values 0 and 1. It follows specific laws like Commutative, Distributive, and Idempotent laws which differ from basic arithmetic.
Step 2: Detailed Explanation:
- Option (A): \( a + b = b + a \). This is the Commutative Law for OR. It is valid.
- Option (B): \( a \cdot (b + c) = (a \cdot b) + (a \cdot c) \). This is the Distributive Law of AND over OR. It is valid.
- Option (C): \( a + a = 2a \). In Boolean algebra, the Idempotent Law states \( a + a = a \). There are no numerical coefficients like \( 2 \) in Boolean logic. This is invalid.
- Option (D): \( a + (b \cdot c) = (a + b) \cdot (a + c) \). This is the Distributive Law of OR over AND. It is valid in Boolean logic, unlike standard algebra.
Step 3: Final Answer:
Option (C) is the incorrect property.