Boolean expression $\sim (A \cdot B) \cdot (\sim A+B) \cdot (\sim B+B)$ is equal to which of the following ?

Question

Accepted Answer

The correct option is (1):
This problem requires simplifying the given Boolean expression using the fundamental laws of Boolean algebra. The initial expression is $E = \sim (A \cdot B) \cdot (\sim A+B) \cdot (\sim B+B)$. We begin by simplifying the rightmost term, $(\sim B+B)$. According to the Complement Law, the OR of a variable and its complement always evaluates to the logical constant $1$ (i.e., $X + \sim X = 1$). Thus, $(\sim B+B)$ simplifies to $1$. The expression then becomes $E = \sim (A \cdot B) \cdot (\sim A+B) \cdot 1$, which further simplifies to $E = \sim (A \cdot B) \cdot (\sim A+B)$ using the Identity Law ($X \cdot 1 = X$). Next, we apply De Morgan's Law to the first term, $\sim (A \cdot B)$, which gives us $(\sim A + \sim B)$. Our expression is now $E = (\sim A + \sim B) \cdot (\sim A+B)$. Finally, we apply the Distributive Law in the form $(X+Y) \cdot (X+Z) = X + (Y \cdot Z)$, where $X = \sim A$, $Y = \sim B$, and $Z = B$. This yields $E = \sim A + (\sim B \cdot B)$. Since the Complement Law states $X \cdot \sim X = 0$, the term $(\sim B \cdot B)$ simplifies to $0$. The final result is $E = \sim A + 0$, which simplifies to $E = \sim A$ by the Identity Law ($X + 0 = X$).

The incorrect option is (2):
The simplified form of the expression is definitively $\sim A$, which is not logically equivalent to $\sim B$. The two terms, $\sim A$ and $\sim B$, are complements of their respective variables, and they will only be equal if $A$ and $B$ have the same value, which is not guaranteed for a general Boolean expression. If the original expression were equal to $\sim B$, the entire process of simplification would need to lead to a dependency only on the variable $B$. The steps clearly show the dependency on $B$ cancels out early, as $(\sim B \cdot B)$ reduces to $0$, isolating $\sim A$ as the final term.

The incorrect option is (3):
The option $\sim A + \sim B$ is equivalent to $\sim (A \cdot B)$ by De Morgan's Law, which was only an intermediate step in the simplification of the original expression. The full expression was $\sim (A \cdot B) \cdot (\sim A+B) \cdot (\sim B+B)$, and it included multiplication by $(\sim A+B)$ and by $1$. Had the final term not been present, the expression would have simplified to $\sim A + \sim B$, but the presence of the other terms allowed for further reduction. Specifically, the factor $(\sim A+B)$ is what enables the second term, $\sim B$, to be eliminated from $\sim A + \sim B$ through the property $X + (\sim X \cdot Y) = X + Y$.

The incorrect option is (4):
The option $A \cdot B$, which represents the logical AND operation, is the opposite of the final simplified result. The simplified expression is $\sim A$. $A \cdot B$ would only be $1$ when both $A$ and $B$ are $1$. In contrast, $\sim A$ is $1$ only when $A$ is $0$, regardless of the value of $B$. Because the final, equivalent form of the original complex expression is $\sim A$, the option $A \cdot B$ cannot be correct as it is logically different. This option is $0$ when the correct answer is $1$ (for instance, when $A=0$ and $B=0$).