GATE CS 1997 Q5.5 — Combinational Circuits

The logic circuit consists of an AND gate followed by an OR gate:

• The inputs to the AND gate are $f_1$ and $f_2$.
• The inputs to the OR gate are the output of the AND gate and $f_3$, producing the final output $f$.

Thus, the boolean expression for the output is:

First, we find the output of the AND gate by taking the intersection of the minterms of $f_1$ and $f_2$:

Now, substituting this into the expression for $f$:

We are given that $f = \sum(8, 9)$. Therefore:

This implies that $f_3$ must contain the minterm $9$ and cannot contain any minterm outside of $\{8, 9\}$. Testing the options, only Option B, which is $\sum(9)$, satisfies this condition perfectly without introducing any unwanted minterms.