Answer to Question #259931 in Discrete Mathematics for thuy

Question

1. Consider the K-Maps given below. For each K- Map

i. Write the appropriate standard form (SOP/POS) of Boolean expression.

ii. Design the circuit using AND, NOT and OR gates.

iii. Design the circuit only by using

• NAND gates if the standard form obtained in part (i) is SOP.

• NOR gates if the standard form obtained in pat (i) is POS.

AB/C

0

1

00

1

0

01

1

1

11

1

0

10

0

1

Answer 1

Part a) is for the left K-map, Part b) is for the right K-map.

a.

i)

$SOP=A'BC+ABC+AB'C'$

$POS=(A'B'+C')(A'B'+C)(A'B+C')(AB+C')(AB'+C)$

$(A'B'+C')=(A'+C')(B'+C')=(A'+C'+B')(A'+C'+B)\times$

$\times (B'+C'+A')(B'+C'+A)=(A'+C'+B')(A'+C'+B))(B'+C'+A)$

$(A'B'+C)=(A'+C)(B'+C)=(A'+C+B)(A'+C+B')\times$

$\times(B'+C+A)(B'+C+A')=(A'+C+B)(A'+C+B')(B'+C+A)$

$(A'B+C')=(A'+C')(B+C')=(A'+C'+B)(A'+C'+B')\times$

$\times(B+C'+A)(B+C'+A')=(A'+C'+B)(A'+C'+B')(B+C'+A)$

$(AB+C')=(A+C')(B+C')=(A+C'+B)(A+C'+B')\times$

$\times(B+C'+A)(B+C'+A')=(A+C'+B)(A+C'+B')(B+C'+A')$

$(AB'+C)=(A+C)(B'+C)=(A+C+B)(A+C+B')\times$

$\times(B'+C+A)(B'+C+A')=(A+C+B')(A+C+B')(B'+C+A')$

Standard form:

$POS=(A'+C'+B')(A'+C'+B))(B'+C'+A)(A'+C+B)(A'+C+B')\times$

$\times(B'+C+A)(B+C'+A)$

ii)

iii)

OR gate using NAND gate:

AND gate using NAND gate:

Then:

b)

I)

$SOP=A'B'C'D'+A'B'CD'+A'BC'D+A'BCD'+ABC'D'+$

$+ABC'D+ABCD+AB'C'D'+AB'C'D+AB'CD+AB'CD'$

$POS=(A'B'+C'D)(A'B'+CD)(A'B+C'D')(A'B+CD)(AB+CD')$

ii)

b)

iii)

SOP using NAND gates:

iii)

POS using NOR gates

realize basic logic gates ( AND, OR, NOT) with NOR gate:

a)

b)