F1 (X,Y,Z) = XY + X'Z

F2(X,Y,Z) = (X+Y).(X'+Z)

What's the relation between F1 & F2 ?

Question

F1 (X,Y,Z) = XY + X'Z

F2(X,Y,Z) = (X+Y).(X'+Z)

What's the relation between F1 & F2 ?

BlackTom AI · Accepted Answer

The problem asks for the relation between two Boolean expressions: F1(X,Y,Z) = XY + X'Z and F2(X,Y,Z) = (X+Y)(X'+Z).

Option 1: Complement. This would mean F2 is the logical complement of F1, i.e., F2 = (XY + X'Z)'. To verify this, one would need F2 to be true exactly when F1 is false. Expanding F2 provides a more direct comparison.

Option 2: Equivalent. Check by algebraic manipulation. Start by expanding F2:
- (X+Y)(X'+Z) = X·X' + X·Z + Y·X' + Y·Z.
- Since X·X' = 0, this reduces to F2 = XZ + X'Y + YZ.
Now apply the consensus theorem: XY + X'Z + YZ = XY + X'Z. Here, if you consider XY + X'Z as the base terms, YZ is the consensus (redundant) term with respect to those two,
so XY + X'Z + YZ simplifies to XY + X'Z. Therefore F2 simplifies to XY + X'Z, which is exactly F1. Hence F1 and F2 are equivalent.

Option 3: Not related. This would imply no logical connection between the two expressions. Given the algebraic expansion shows that F2 reduces to the same form as F1, this option is incorrect.

Option 4: Dual. In Boolean logic, a dual swaps AND/OR and 0/1 constants. There is no indication that F2 is the dual of F1; expanding F2 does not yield a simple dual of F1. Therefore this option does not describe their relationship.

Summarizing the expansions and simplification:
- F2 = (X+Y)(X'+Z) = XZ + X'Y + YZ.
- By the consensus theorem, XZ + X'Y + YZ = XY + X'Z.
- Thus F2 simplifies to F1, so the two functions are equivalent.

This analysis shows that the only statement describing their relationship accurately is that the two functions are equivalent.

F1 (X,Y,Z) = XY + X'Z F2(X,Y,Z) = (X+Y).(X'+Z) What's the relation between F1 & F2 ?

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