Consider the Boolean function $f=(a+bc).(pq+r)$ Complement $f'$ of function f is :

Question

Accepted Answer

The correct option is (2):
The function $f=(a+bc).(pq+r)$ is a conjunction (AND operation) of two large terms: $A=(a+bc)$ and $B=(pq+r)$. To find the complement $f'$, we must apply De Morgan's laws. De Morgan's Second Law, which states that the complement of a product is the sum of the complements ($\overline{A \cdot B} = \overline{A} + \overline{B}$), is applied first. This gives $f' = \overline{(a+bc)} + \overline{(pq+r)}$. Next, we apply De Morgan's First Law ($\overline{A+B} = \overline{A} \cdot \overline{B}$) to the two new terms. $\overline{(a+bc)}$ becomes $\overline{a} \cdot \overline{(bc)}$, and then $\overline{(bc)}$ is resolved as $\overline{b}+\overline{c}$, yielding $a'(b'+c')$. Similarly, $\overline{(pq+r)}$ becomes $\overline{(pq)} \cdot \overline{r}$, and $\overline{(pq)}$ is resolved as $\overline{p}+\overline{q}$, yielding $(p'+q')r'$. Combining these two results, we get the fully simplified complement: $f' = a'(b'+c') + (p'+q')r'$.

The incorrect option is (1):
Option 1, $(a'+b'c').(p'q'+r')$, is incorrect because it fails to correctly apply the outermost De Morgan's Law. The primary operation in the original function is AND, and applying the complement must convert this to an OR operation, as per the law $\overline{A \cdot B} = \overline{A} + \overline{B}$. This option incorrectly maintains the outermost AND operation (product form) between the two main groups. Furthermore, the inner application to $\overline{(bc)}$ is incorrectly written as $b'c'$ when De Morgan's law would require it to be $b'+c'$ (sum of complements), indicating compounding errors in the application of the complement rules.

The incorrect option is (3):
Option 3, $(a'+b'c')+(p'q'+r')$, correctly inverts the outermost AND to an OR, which is a good first step. However, it fails in the internal application of De Morgan's First Law for terms that are sums. The complement of the sum $\overline{(a+bc)}$ should resolve to a product of complemented terms: $\overline{a} \cdot \overline{(bc)}$, which is $a' \cdot (b'+c')$. This option incorrectly keeps the OR structure inside the group, writing $a'+b'c'$ instead of the required product form, $a'(b'+c')$. The same error is present in the second group, where $\overline{(pq+r)}$ is incorrectly shown as $p'q'+r'$.

The incorrect option is (4):
Option 4, $(a'b'c')+(p'q'r')$, makes a fundamental mistake in distributing the complement over a sum. When calculating $\overline{(a+bc)}$, the rule $\overline{A+B} = \overline{A} \cdot \overline{B}$ must be applied hierarchically. Instead, this option treats $a$, $b$, and $c$ as if they were a product under the complementation bar or applies De Morgan's law multiple times incorrectly without proper grouping. The correct expansion requires the complement of the sum $a+bc$ to result in the product $a'(b'+c')$. This option simplifies the terms far too aggressively into a product of all three complemented variables $a'b'c'$, demonstrating a misapplication of the complement over the original OR operations.