Minimize the following function using Quine McCluskey method: f(A,B,C,D)=Πm(2,7,8,9,10,12)

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teamques10 ★ 64k

Quine McCluskey Method is a tabular method of minimisation.

Here maxterms are given:

So, the rest of the numericals between 0 to 15 are minterms.

Use the minterms. Find the minimized function in Sum of products (SOP).

Apply De Morgan's Law to SOP to get product of sums(POS).

$f(A,B,C,D)= \sum m(0,1,3,4,5,6,11,13,15)$

Step 1: Input Grouping the minterms/don't care terms based on number of 1's.

Index	Minterm/Don't care terms	Binary Representation
0	0	0	0	0	0
1	1	0	0	0	1
4	0	1	0	0
2	3	0	0	1	1
5	0	1	0	1
6	0	1	1	0
3	11	1	0	1	1
13	1	1	0	1
14	1	1	1	0
4	15	1	1	1	1

Step 2: First Comparison Group the terms in pairs:

Group	Pairs	Binary Representation
0	(1,0)	0	0	0	-
1	(4,0)	0	-	0	0
(3,1)	0	0	-	1
(5,1)	0	-	0	1
(5,4)	0	1	0	-
(6,4)	0	1	-	0
2	(11,3)	-	0	1	1
(13,5)	-	1	0	1
(14,6)	-	1	1	0
3	(15,11)	1	-	1	1
(15,13)	1	1	-	1
(15,14)	1	1	1	-

Step 3: Second Comparison

Group	Pairs	Binary Representation
0	(5,4,1,0)	0	-	0	-

Step 4: Prime Implicants

(5,4,1,0) 0 - 0 -

(3,1) 0 0 - 1

(6,4) 0 1 - 0

(11,3) - 0 1 1

(13,5) - 1 0 1

(14,6) - 1 1 0

(15,11) 1 - 1 1

(15,13) 1 1 - 1

(15,14) 1 1 1 -

Step 4:Coverage Table

	0-0-	00-1	01-0	-011	-101	-110	1-11	11-1	111-
0	x
1	x	x
3		x		x
4	x		x
5	x				x
6			x			x
11				x			x
13					x			x
14						x			x
15							x	x	x

$f=A'C'+ABD+B'CD+BCD'\\ \text{Applying De Morgan's law to find POS}\\ f'=(A'C')' . (ABD)'.(B'CD)'(BCD')'\\ f'=[(A')'+(C')'][A'+B'+D'][(B')'+C'+D'][B'+C'+(D')']\\ f'=[A+C][A'+B'+D'][B+C'+D'][B'+C'+D]$

Verify the output using this handy tool.

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