Boolean function (X + Y + Z).(X + Y + Z').(X + Z') can be minimized to

Concept

Two product-of-sums factors that agree in every literal except one variable that appears complemented in one and uncomplemented in the other combine by the rule (A + B)(A + B') = A. Distribution then gives (X + A)(X + B) = X + A.B. The rule A + A.B = A (absorption) only removes a product term that already contains A as a factor.

Application

1. Group the first two factors, which share (X + Y) and differ only in Z vs Z': (X + Y + Z)(X + Y + Z') = (X + Y), by (A + B)(A + B') = A with A = X + Y, B = Z.

2. The expression becomes (X + Y)(X + Z').

3. Apply (X + A)(X + B) = X + A.B with A = Y, B = Z': (X + Y)(X + Z') = X + Y.Z'.

4. The term Y.Z' is not absorbed in X + Y.Z': absorption would require a standalone summand that is also a factor of the product term, as in Y + Y.Z' = Y. Here the other summand is X, and Y.Z' does not contain X as a factor, so the product term remains.

So the minimized sum-of-products form is X + Y.Z'.

Cross-check

Truth-table check: setting X = 0 collapses the original product to (Y + Z)(Y + Z')(Z') = Y.Z' (the same (A + B)(A + B') = A rule, with A = Y, B = Z), so it is 1 only when Y = 1 and Z = 0, and 0 at the other three X = 0 rows. At Y = 1, Z = 1 this gives Y.Z' = 1.0 = 0, and X + Y.Z' reproduces that same 0, while X + Y would give 0 + 1 = 1 -- so X + Y fails to match the function there, whereas X + Y.Z' matches it at all eight input combinations.

Final result

Therefore, (X + Y + Z)(X + Y + Z')(X + Z') minimizes to X + Y.Z'.