Truth table of Boolean function X.Y.Z + X.Y.Z' + X.Y.Z + X'.Y matches with
2013
Truth table of Boolean function X.Y.Z + X.Y.Z' + X.Y.Z + X'.Y matches with
Answer: B. Y — ConceptA sum-of-products expression simplifies by two Boolean identities. Combining: a variable ANDed with both a literal and its complement drops that…
- A.
(X ∧ Y)'
- B.
Y
- C.
Z
- D.
None of these
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Correct answer: B
Concept
A sum-of-products expression simplifies by two Boolean identities. Combining: a variable ANDed with both a literal and its complement drops that literal, X.Y.Z + X.Y.Z' = X.Y.(Z + Z') = X.Y, because Z + Z' = 1. Distribution in reverse: a common factor ANDed with a variable and its complement leaves only the common factor, since X + X' = 1.
Application
Simplify X.Y.Z + X.Y.Z' + X.Y.Z + X'.Y step by step:
By idempotency (A + A = A) the repeated term X.Y.Z is absorbed, leaving X.Y.Z + X.Y.Z' + X'.Y.
Factor the first two terms: X.Y.Z + X.Y.Z' = X.Y.(Z + Z'). Since Z + Z' = 1, this is X.Y.
The expression is now X.Y + X'.Y. Factor out Y: (X + X').Y.
Since X + X' = 1, this gives 1.Y = Y.
Cross-check
Build the truth table. The expression is 1 exactly on the rows where Y = 1 (X,Y,Z = 010, 011, 110, 111) and 0 wherever Y = 0, independent of X and Z. That column is identical to the column for Y, confirming the function equals Y.