Construct a truth table for the following Boolean expressions. $[15$ Marks$]$

$A.(A+B^{\text{'}}+C)(A^{\text{'}}+B+C^{\text{'}})$

B$.$

C$.(AB)+(A^{\text{'}}{B}^{\text{'}})+(A{C}^{\text{'}})$

A particular brand of automatic garage door opener utilizes a transmitter control with $3$ independent

switches, $A,B,$ and $C,$ each one set $\text{'}$on $(1)\text{'}$ or 'off $(0)\text{'}.$ The garage door will be opened $(x=1)$ if at least one of

the switches is $\text{'}$on$.\text{'}[10$ Marks$]$

A$.$ Construct a truth table for the above information. And obtain a boolean expression for the outputs

of 'open the door.

B$.$ Simplify the SOP expression using K$-$Map.

C$.$ Draw the circuit diagram using gates.

Write SOP and Simplify the following Boolean expression using a Karnaugh map and draw a circuit diagram.

$[40$ Marks$]$

A$.F(A,B,C,D)=(1,2,5,7,9,11,13,15)$

B$.F(A,B,C,D)=(0,1,2,4,5,6,8,9,10,11)$

C$.F(A,B,C,D)=(1,2,6,8,10,15)$

D$.F(A,B,C,D)=(0,1,2,3,6,8,9,10,11,14)$

Construct the AND, OR$,$ and NOT gates from the basic Boolean operations NOR and NAND. $[10$ Marks$]$