Given the Boolean function . Reduce the above expression by using a 4-variable Karnaugh map…
Computer Science20255 marksShort answer
Given the Boolean function $F(P,Q,R,S) = \Sigma(2,3,5,7,8,10,11,12,13,15)$.
(a)[3.0]
Reduce the above expression by using a 4-variable Karnaugh map, showing the various groups (i.e. octal, quads and pairs).
(b)[2.0]
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
Answer
Answer (a)
AI
K-map for $F(P,Q,R,S) = \Sigma(2,3,5,7,8,10,11,12,13,15)$ (verified with the boolean/K-map tool):
Pair ($m_8,m_{12}$): P=1, R=0, S=0 -> term $PR'S'$
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 $Q'R$
Reduced (minimal) SOP expression: $F(P,Q,R,S) = PR'S' + QS + Q'R$
Answer (b)
AI
NAND-only (NAND-NAND) implementation of $F = PR'S' + QS + Q'R$: one NAND gate for each of the three product terms, and a 3-input NAND gate combining them, since $[(PR'S')' \cdot (QS)' \cdot (Q'R)']' = PR'S' + QS + Q'R$ by De Morgan's law. Variables and their complements (P, P', Q, Q', R, R', S, S') are assumed available as inputs, as stated.
