The simultaneous equations on the Boolean variables x, y, z and w are: x + y +…
2000
The simultaneous equations on the Boolean variables x, y, z and w are:
x + y + z = 1
xy = 0
xz + w = 1
xy + z'w' = 0
They have the following solution for x, y, z and w, respectively.
Answer: C. 1 0 1 1 — CONCEPTIn Boolean algebra, + denotes OR, adjacency denotes AND, and an apostrophe denotes complement. Thus 1 + 0 = 1, 1·0 = 0, and if z = 1 then z' = 0. A…
- A.
0 1 0 0
- B.
1 1 0 1
- C.
1 0 1 1
- D.
1 0 0 0
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Correct answer: C
CONCEPT
In Boolean algebra, + denotes OR, adjacency denotes AND, and an apostrophe denotes complement. Thus 1 + 0 = 1, 1·0 = 0, and if z = 1 then z' = 0.
A simultaneous assignment must make every left-hand side equal its stated right-hand side. The reliable method is to substitute one tuple and evaluate the equations in order.
APPLICATION
For x = 1, y = 0, z = 1, w = 1, the first left-hand side is x + y + z = 1 + 0 + 1 = 1 under Boolean OR.
The second left-hand side is xy = 1·0 = 0.
The third left-hand side is xz + w = 1·1 + 1 = 1.
Here z' = 0 and w' = 0, so the fourth left-hand side is xy + z'w' = 1·0 + 0·0 = 0.
CROSS-CHECK / CONTRAST
For (0, 1, 0, 0), xz + w evaluates to 0 and xy + z'w' evaluates to 1.
For (1, 1, 0, 1), xy evaluates to 1 and xy + z'w' evaluates to 1.
For (1, 0, 0, 0), xz + w evaluates to 0 and xy + z'w' evaluates to 1.
Only the offered tuple x = 1, y = 0, z = 1, w = 1 produces the required sequence 1, 0, 1, 0 for the four equations. Therefore, that tuple is the answer.