Logic gates and circuit simplification (intro) - Discrete Math

1. A circuit designer claims that the circuit implementing $f(x,y,z) = xyz + x\overline{y}z + \overline{x}yz + \overline{x}\overline{y}z$ can be simplified to just a single gate. Which single gate type correctly implements this function?
   Answer: A buffer gate with input z, as all terms contain z and the remaining expression equals 1 Factoring out z: f(x,y,z) = z(xy + xȳ + x̄y + x̄ȳ). The expression in parentheses covers all possible combinations of x and y (00, 01, 10, 11), so it equals 1 for all inputs. Therefore f = z·1 = z. A buffer gate simply passes its input unchanged. Choice A describes a wire connection rather than a gate, B gives wrong simplification, D ignores the z dependency.
2. Two equivalent circuits implement the same Boolean function. Circuit A uses 8 two-input NAND gates, while Circuit B uses 3 two-input NOR gates and 2 inverters. Assuming identical propagation delays for all gate types, which statement about power consumption is most accurate?
   Answer: Power consumption depends on input patterns and cannot be determined from gate counts alone Power consumption in CMOS logic depends heavily on switching activity, input transition frequencies, and the specific input patterns applied to the gates. Gate count alone is insufficient to determine power consumption. While Circuit A has more gates, the actual power depends on how often each gate switches, which varies with input patterns. Choices A, B, and C make assumptions about power based solely on gate types or counts.
3. A logic circuit implements the function $f(x,y,z) = xy + \overline{x}z + yz$. After applying Boolean algebra simplification, which expression represents the most reduced form?
   Answer: $xy + \overline{x}z$, since the consensus term yz is redundant by the consensus theorem The consensus theorem states that AB + ĀC + BC = AB + ĀC, where the term BC is redundant. Here, xy corresponds to AB, x̄z corresponds to ĀC, and yz corresponds to BC. Therefore, yz can be eliminated, giving xy + x̄z. Choice B is factored but not minimal, C is incorrectly distributed, and D incorrectly eliminates xy.
4. A multiplexer circuit has the truth table behavior where output Y equals input D₀ when select lines S₁S₀ = 00, equals D₁ when S₁S₀ = 01, equals D₂ when S₁S₀ = 10, and equals D₃ when S₁S₀ = 11. What Boolean expression represents this multiplexer function?
   Answer: $Y = \overline{S_1}\overline{S_0}D_0 + \overline{S_1}S_0D_1 + S_1\overline{S_0}D_2 + S_1S_0D_3$ A multiplexer selects one input based on select lines. Each data input is gated by the AND of the select lines in the appropriate combination: D₀ is selected when both S₁ and S₀ are 0 (S̄₁S̄₀), D₁ when S₁=0 and S₀=1 (S̄₁S₀), etc. Choice B reverses the select conditions, C uses OR instead of AND, and D incorrectly applies De Morgan's law to S₁S₀.

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