Consider a Boolean expression given by 𝐹(𝑋, 𝑌, 𝑍) = ∑(3,5,6,7). Which of…

Solution:

Given F(X, Y, Z) = Σ(3,5,6,7).

• Minterms where F = 1 are indices 3, 5, 6, 7 corresponding to binary inputs 011, 101, 110, 111.

• Therefore the maxterms (where F = 0) are the remaining indices 0, 1, 2, 4, so F can be written as the product of maxterms Π(0,1,2,4).

Derive a simplified sum-of-products form from the minterms:

1. Write the minterms: X' Y Z + X Y' Z + X Y Z' + X Y Z.

2. Combine terms: X Y Z' + X Y Z = X Y (Z' + Z) = X Y.

3. Also group the remaining terms with Z: X' Y Z + X Y' Z = Z (X' Y + X Y') = Z (X ⊕ Y) , and together with XY this yields the symmetric form XY + XZ + YZ.

Thus the simplified expression is F = XY + XZ + YZ.

Conclusion about the given statements:

• The statement "F(X, Y, Z) = Π(0,1,2,4)" is correct because those indices are exactly where F = 0.

• The statement "F(X, Y, Z) = XY + YZ + XZ" is correct; this is the simplified sum-of-products form.

• The claims that F is independent of Y or independent of X are both incorrect. Counterexamples:

• Dependence on Y: With X=1, Z=0, Y=0 gives input 100 (index 4) → F=0, while Y=1 gives input 110 (index 6) → F=1, so changing Y changes F.

• Dependence on X: With Y=0, Z=1, X=0 gives input 001 (index 1) → F=0, while X=1 gives input 101 (index 5) → F=1, so changing X changes F.