Digital Design I

HW#$2$

For the Boolean function

$F=x{y}^{\text{'}}z+x^{\text{'}}{y}^{\text{'}}z+w^{\text{'}}xy+w{x}^{\text{'}}y+wxy$

$($a$)$ Obtain the truth table of $F.$

$($b$)$ Draw the logic diagram, using the original Boolean expression.

$($c$)$ Use Boolean algebra to simplify the function to a minimum number of literals.

$($d$)$ Draw the logic diagram from the simplified expression, and compare the total

number of gates with the diagram of part $($b$).$

Draw the logic diagram corresponding to the following Boolean expressions

without simplifying them:

$($a$)Y=[(u+x^{\text{'}})(y^{\text{'}}+z)]$

$($b$)Y=u(xo+z)+y^{\text{'}}$

$($c$)Y=u+x+x^{\text{'}}(u+y^{\text{'}})$

Simplify the following expressions to $(1)$ sum$-$of$-$products and $(2)$ products$-$of$-$

sums:

$($a$)F=x^{\text{'}}{z}^{\text{'}}+y^{\text{'}}{z}^{\text{'}}+y{z}^{\text{'}}+xy$

$($b$)F=(A+B+D^{\text{'}})(A^{\text{'}}+B^{\text{'}}+C^{\text{'}})(A^{\text{'}}+B^{\text{'}}+C)(B^{\text{'}}+C+D^{\text{'}})$

Simplify the following Boolean functions $F,$ together with the don't$-$care conditions

$d,$ and then express the simplified function in sum$-$of$-$minterms form:

$($a$)F(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(2,3,4,6,7)$

$($b$)F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(4,5,7,12,13,14)$

$d(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}(0,1,5)$

$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(1,9,11,15)$

$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(0,2,9)$

$($c$)F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(1,3,8,10,15)$

$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(1,9,11,15)$

$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(0,2,9)$

Draw a NAND logic diagram that implements the complement of the following

function: $F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(0,1,2,3,6,10,11,14).$