If \(w, x, y, z\) are Boolean variables, then which one of the following is…

Solution: simplify each given Boolean expression using standard identities (distribution, De Morgan, absorption) and check whether the two sides are equal.

• For wx + w(x+y) + x(x+y) = x + wy: w(x+y) = wx + wy and x(x+y) = x, so the left side becomes wx + (wx + wy) + x = wx + wy + x. Using absorption x + wx = x, this simplifies to x + wy, matching the right side.

• For not(w · not x · (y + not z)) + not w · x = not w + x + not y · z: apply De Morgan to the negation to get (not w) OR x OR (not y AND z). The additional term (not w AND x) is redundant because it is already covered by (not w) OR x, so the whole expression simplifies to not w + x + not y · z, matching the right side.

• For (w · not x · (y + x · not z) + not w · not x) · y = x · not y: factor not x from the parentheses to get not x · (not w + y + x · not z). Multiplying by y gives not x · y. Therefore the left side simplifies to not x AND y, while the right side is x AND not y. These are not equal in general. For example, with x = 0 and y = 1, left = 1 but right = 0. Hence this equality is false.

• For (w + y) · (wxy + wyz) = wxy + wyz: factor w y from the second factor to get wy(x + z). Then (w + y)·wy(x + z) = wy(x + z) because (w + y)·wy = wy by absorption, which equals wxy + wyz. So the equality holds.

Conclusion: the third given expression (the one that simplifies to not x · y on the left but to x · not y on the right) is incorrect; the others are correct.