Which of the following is not a valid rule of Boolean Algebra ?

Question

Accepted Answer

The correct option is (1):
The expression $\text{A+A'}=\text{A}$ is the one that is **not a valid rule** in Boolean Algebra. This expression incorrectly attempts to formulate the logical OR operation with a variable and its complement. The correct and fundamental law in Boolean Algebra is the **Complementarity Law** for OR, which correctly states that $\text{A+A'}=1$. To prove why $\text{A+A'}=\text{A}$ is invalid, consider a case: if the variable $\text{A}$ is 0, the equation suggests $0+0'=0$. However, substituting the values gives $0+1=1$, and since $1$ does not equal $0$, the rule is false. Therefore, $\text{A+A'}=1$ is the correct identity, making $\text{A+A'}=\text{A}$ the invalid rule among the options.

The incorrect option is (2):
The rule $\text{A}.\text{A}=\text{A}$ is a **valid** and essential law of Boolean Algebra known as the **Idempotent Law** for the AND operation. This law states that when a Boolean variable is ANDed with itself, the result is the original variable. This holds true whether $\text{A}$ is 0 ($0.0=0$) or $\text{A}$ is 1 ($1.1=1$). This property is frequently used to simplify Boolean expressions by removing redundant identical terms.

The incorrect option is (3):
The expression $\text{A}.\text{A'}=0$ is a **valid** and critical rule, representing the second part of the **Complementarity Law** for the AND operation. This law dictates that a Boolean variable ANDed with its complement will always yield 0, which represents the null set. Since $\text{A}$ and $\text{A'}$ must always have opposite values (one is 1, the other is 0), their logical product will inevitably be $1.0$ or $0.1$, which results in 0. This law is foundational in minimizing and simplifying logic circuits.

The incorrect option is (4):
The rule $\text{A+A'}=1$ is a **valid** and fundamental law, correctly stating the **Complementarity Law** for the OR operation. This law indicates that ORing a variable with its complement always produces a result of 1, regardless of the variable's value. Since the OR operation is true (1) if at least one input is true, and since either $\text{A}$ or $\text{A'}$ must always be 1, the output is guaranteed to be 1. This identity is a cornerstone of digital logic and theorem proving.