Given the Boolean function F(P,Q,R,S) = Σ(2,3,5,7,8,10,11,12,13,15). Reduce the above expression by using a 4-variable Karnaugh map, showing the various…

Quad (m₅, m₇, m₁₃, m₁₅): Q = 1, S = 1, term QS Quad (m₂, m₃, m₁₀, m₁₁): Q = 0, R = 1, term Q'R Pair (m₈, m₁₂): P = 1, R = 0, S = 0, term PR'S' Hence F(P,Q,R,S) = QS + Q'R + PR'S'

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Given the Boolean function F(P,Q,R,S) = Σ(2,3,5,7,8,10,11,12,13,15). Reduce the above expression by…

Given the Boolean function $F(P,Q,R,S) = \Sigma(2,3,5,7,8,10,11,12,13,15)$.

(a)[3]

Reduce the above expression by using a 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).

(b)[2]

Draw the logic gate diagram for the reduced expression using NAND gate only. Assume that the variables and their complements are available as inputs.

Draw: Logic gate diagram for reduced expression using only NAND gates

Must show: NAND gates

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Answer

Answer (a)

Groups: (5, 7, 13, 15) → QS; (2, 3, 10, 11) → Q'R; (8, 12) → PR'S'

Reduced expression: F = QS + Q'R + PR'S'

Quad ($m_5, m_7, m_{13}, m_{15}$): $Q = 1, S = 1$, term $QS$ Quad ($m_2, m_3, m_{10}, m_{11}$): $Q = 0, R = 1$, term $Q'R$ Pair ($m_8, m_{12}$): $P = 1, R = 0, S = 0$, term $PR'S'$ Hence $F(P,Q,R,S) = QS + Q'R + PR'S'$

Answer (b)

NAND-NAND form of $F = QS + Q'R + PR'S'$: one NAND gate for each product term and a 3-input NAND gate combining them, since $[(QS)' \cdot (Q'R)' \cdot (PR'S')']' = QS + Q'R + PR'S'$ by De Morgan's law.