F(P, Q, R, S) = Σ (0,1,2,5,7,8,9,10,13,15) Reduce the Boolean function F(P, Q, R, S) = Σ (0,1,2,5,7,8,9,10,13,15) by using 4-variable Karnaugh map…

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Reduce the following using a Karnaugh map.

F(P, Q, R, S) = Σ (0,1,2,5,7,8,9,10,13,15) Reduce the Boolean function F(P, Q, R, S) = Σ…

$F(P, Q, R, S) = \Sigma (0,1,2,5,7,8,9,10,13,15)$

(a)[4]

Reduce the Boolean function $F(P, Q, R, S) = \Sigma (0,1,2,5,7,8,9,10,13,15)$ by using 4-variable Karnaugh map, showing the various groups (i.e., octal, quads and pairs).

(b)[1]

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

Draw: logic gate diagram using NAND gates only for the reduced expression

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Answer

Answer (a)

Groups: (5, 7, 13, 15) → QS; (0, 1, 8, 9) → Q'R'; (0, 2, 8, 10) → Q'S'

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

Quad (5, 7, 13, 15) gives QS. Quad (0, 1, 8, 9) gives Q'R'. Quad (0, 2, 8, 10) gives Q'S'. There is no octet.

Answer (b)

NAND-only (NAND-NAND) implementation of $F = QS + Q'R' + Q'S'$: one 2-input NAND gate for each product term and a 3-input NAND gate combining them, since $[(QS)' \cdot (Q'R')' \cdot (Q'S')']' = QS + Q'R' + Q'S'$ by De Morgan's law.