GATE CSE 2024 | Set 1 | Question: 37

Question

Consider a Boolean expression given by $\text{F(X, Y, Z)}=\sum(3,5,6,7)$.

Which of the following statements is/are CORRECT?

• A. $\text{F(X, Y, Z)}=\Pi(0,1,2,4)$ 
• B. $\text{F(X, Y, Z)=X Y+Y Z+X Z}$
• C. $\text{F(X, Y, Z)}$ is independent of input $\text{Y}$
• D. $\text{F(X, Y, Z)}$ is independent of input $\text{X}$

Comments

When the min-term is given then indirectly max-term is also given which is not part of the min-term.

min-term - 1 in Karnaugh map

max-term - 0 in Karnaugh map.

Answer 1

$\implies F(X, Y, Z) = \prod(0, 1, 2, 4)$

$\implies F(X, Y, Z) = XY + XZ + YZ$

This implies F(X, Y, Z) is dependent on both X and Y.

Answer - A, B

Answer 2

The given boolean function $f(x,y,z)=\sum_m(3,5,6,7)$ is in sop form (minterm),

 so its equivalent POS (maxterm) expression contains all the terms that are not present in the terms of three variables.

That is $f(x,y,z)=\Pi_m(0,1,2,4)$ is correct.

Now $f(x,y,z)=\sum_m(3,5,6,7)$ when we minimize this using kmap we get $(xy+yz+zx)$ as expression.

So given that minimized expression is not independent from $x,y,z$

Option $(A,B)$ are correct.