Consider the Boolean expression given by F=(X+Y+Z)(X'+Y)(Y'+Z). Which of the following Boolean expressions is/are equivalent to F' (Complement of F)? A. (X'+Y'+Z')(X+Y')(Y+Z') B. (XY'+Z') C. (X+Z')(Y'+Z') D. (XY'+YZ'+X'Y'Z') Choose the correct answer from the options given below:
B, C and D Only
Two Boolean expressions are equivalent when they take the value 1 for exactly the same input combinations, so the quickest check is to list the minterms (XYZ read as a 3-bit number) of F, of F' and of each candidate.
Step 1 — F = 1 needs all three sums to be 1. If Y = 0, (X'+Y) forces X = 0 and (X+Y+Z) forces Z = 1, giving 001. If Y = 1, (Y'+Z) forces Z = 1 and X is free, giving 011 and 111. So F = Σm(1, 3, 7) and F' = Σm(0, 2, 4, 5, 6).
Step 2 — A: the product is 0 at 111 (first sum), at 010 and 011 (X+Y'), and at 001 and 101 (Y+Z'). Its 1s are only Σm(0, 4, 6), so A is not F'.
Step 3 — B: XY' covers 100, 101 and Z' covers 000, 010, 100, 110, giving Σm(0, 2, 4, 5, 6) = F'.
Step 4 — C: (X+Z') is 0 at 001, 011 and (Y'+Z') is 0 at 011, 111, so its 0s are Σm(1, 3, 7) and its 1s are Σm(0, 2, 4, 5, 6) = F'.
Step 5 — D: XY' gives 4, 5; YZ' gives 2, 6; X'Y'Z' gives 0. The union is Σm(0, 2, 4, 5, 6) = F'.
B, C and D equal F' while A does not, so every combination that includes A fails.
B, C and D Only option is correct.
Similar Questions
Which of the following is the complement of the boolean function.
A'B+CD'+A'B+CD'
Match the LIST-I with LIST-II
LIST-I
Boolean Algebra LawLIST-II
AxiomsA. Absorption Law I. a+1=1 B. Bounded Law II. a+0=a C. Identity Law III. a*(b+c)=(a*b)+(a*c) D. Distributive Law IV. a+(a*b)=a Choose the correct answer from the options given below:
Important Questions from Logic Gates and Boolean Algebra
In Boolean algebra, the term sum of products means
The value of \(\rm \overline{A+B}\) is :
- If A + B = A + C and AB = AC, then which of the following is true?
The Boolean function Y = AB + CD is to be realized using only two-input NAND gates. The minimum number of gates required are:
The minimum number of 2-input NAND gates required to realize the logic function $Y = AB + \bar A \bar B$ is
