Simplify the following Boolean functions using $3-$variable K$-$maps. In each case, also

identify all PIs and EPIs.

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

b$)F(x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(0,1,2,3,5)$

Simplify the following Boolean functions using $4-$variable K$-$maps. In each case, also

identify all PIs and EPIs.

a$)C^{\text{'}}D+AB{C}^{\text{'}}+AB{D}^{\text{'}}+A^{\text{'}}{B}^{\text{'}}D$

b$)A^{\text{'}}B+A^{\text{'}}CD+B^{\text{'}}CD+B{C}^{\text{'}}{D}^{\text{'}}$

Simplify $($d represents don't cares$)$:

a$)F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(0,6,10,13,14)$;$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(2,5,7,8,11,15)$

b$)F(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(3,8,10,11,15)$;$d(A,B,C,D)={\displaystyle \underset{?}{\overset{?}{}}}(1,2,4,6,9)$

Find the minimum SOP and POS expressions for each of the following functions. Do this

directly from the K$-$map representation of the functions.

a$)F(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(0,2,4,6,8,9,10)$

b$)F(w,x,y,z)=$prodM$(1,2,5,7,9,10,13,14,15)$

Implement the Boolean function $F=xy+x^{\text{'}}{y}^{\text{'}}+y^{\text{'}}z$ using a minimal number of gates. For

each implementation write the minimal logic expression and draw the circuit.

a$)$ with AND, OR$,$ NOT gates only.

b$)$ with NAND, NOT gates only $($note that this could be done with NANDs only$).$

c$)$ with NOR, NOT gates only $($note that this could be done with NORs only$).$

Implement the Boolean function $F(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(3,4,5,11,12,13,14,15)$ using a

minimum number of NAND gates only. Write the minimal logic expression $($no need to draw

the circuit$).$

Implement the Boolean function $F(w,x,y,z)={\displaystyle \underset{?}{\overset{?}{}}}m(3,4,5,11,12,13,14,15)$ using a

minimum number of NOR gates only. Write the minimal logic expression $($no need to draw

the circuit$).$

Design a combinational circuit with three inputs and one output for each of the two cases

below. Show your truth table, K$-$maps, and minimized logic expressions.

a$)$ The output is $1$ when the binary value of the inputs is less than $3,$ and $0$ otherwise.

b$)$ The output is $1$ when the binary value of the inputs is an odd number.