From the logic circuit diagram given below, name the outputs (1), (2) and (3) and finally derive…
Computer Science20244 marksDerivation
From the logic circuit diagram given below, name the outputs (1), (2) and (3) and finally derive the Boolean expression (F) and simplify it. Identify the propositional connective which is equivalent to the simplified Boolean expression.
Answer
Answer
AI
Outputs: (1) = X+Y', (2) = X.Z, (3) = (X+Y').(X.Z). Simplifying, F(X,Y,Z) = (3) + Z' = X + Z', which is logically equivalent to the propositional connective IMPLICATION, $Z \Rightarrow X$.
(1) = X + Y' - output of the top OR gate, combining input X with Y' (from the NOT gate on Y).
(2) = X . Z - output of the AND gate, combining input X (same branch that feeds gate (1)) with input Z.
(3) = (1).(2) = (X+Y').(X.Z) - output of the middle AND gate, combining outputs (1) and (2).
By the Distributive Law: (X+Y').(X.Z) = X.Z.X + X.Z.Y' = X.Z + X.Y'.Z (using the Idempotent Law X.X=X).
X.Z + X.Y'.Z = X.Z.(1+Y') = X.Z (by the Distributive Law and the Identity Law 1+Y'=1); equivalently, by the Absorption Law X.(X+Y')=X, so (3) = X.Z directly.
F = (3) + Z' - output of the final OR gate, combining (3) with Z' (from the bottom NOT gate on Z): F = X.Z + Z'.
By the identity A + A'.B = A + B (with A=Z', B=X): Z' + Z.X = Z' + X.
Therefore F(X,Y,Z) = X + Z'.
X + Z' is equivalent to the propositional connective Implication, Z => X, since P=>Q is equivalent to (~P v Q); here Z=>X is equivalent to Z'+X = X+Z'.
