Given below are four equations. Which one of them is not true? (A) XY + X’Z +…
Given below are four equations. Which one of them is not true?
(A) XY + X’Z + YZ = XY + X’Z
(B) AB + B’C + AC = (B + C) (B’ + A)
(C) (X + Y + Z)’ = X’ + Y’ + Z’
(D) XY + X’Y = Y
Answer: D. C — Answer: As printed, the second equation and the third equation are not true; the first and fourth equations are true. XY + X'Z + YZ = XY + X'Z — True. By the…
- A.
A and B
- B.
D
- C.
B
- D.
C
Attempted by 49 students.
Show answer & explanation
Correct answer: D
Answer: As printed, the second equation and the third equation are not true; the first and fourth equations are true.
XY + X'Z + YZ = XY + X'Z — True. By the consensus theorem AB + A'C + BC = AB + A'C; mapping A=X, B=Y, C=Z shows the term YZ is redundant.
AB + B' C + AC = (B + C)(B' + A)(C) — Not true as printed. The left-hand side simplifies (by the consensus theorem) to AB + B' C because AC is the consensus term and is redundant. The right-hand side simplifies to C(B' + A) = AC + B' C, which omits AB. For example, with A=1, B=1, C=0 the left-hand side is 1 while the right-hand side is 0, so the equality fails.
(X + Y + Z)' = X' + Y' + Z' — Not true. De Morgan's law gives (X + Y + Z)' = X'·Y'·Z' (the AND of complements). For example, X=1,Y=0,Z=0 makes the left-hand side 0, while the right-hand side X' + Y' + Z' equals 1, so the printed equality is false.
XY + X'Y = Y — True. Factor Y: XY + X'Y = Y(X + X') = Y·1 = Y.
Conclusion: As written, both the second printed equality and the De Morgan equality (the third) are incorrect. If the original question expected a single incorrect equation, the second equation likely contains a typographical error on the right-hand side; otherwise both are not true.