(Solved): 1. Simplify the following Boolean expressions using algebraic manipulation. Note: You must show all ...

1. Simplify the following Boolean expressions using algebraic manipulation. Note: You must show all the steps with explanation (laws used) to get the full point. Solutions without the intermediate steps will not be accepted. Note that ' means -not- (i.e., the same as a bar over a variable or expression) Example: $\mathrm{F}(\mathrm{a},\mathrm{b})=\mathrm{b}+(\mathrm{ab}{)}^{?}$ Solution: $\begin{array}{ll}b+(ab{)}^{?}& \\ b+\left(a{b}^{?}+b^{?}\right)& \text{ DeMorgan’s Law }\\ \left(b_{+}{b}^{?}\right)+a^{?})& \text{ Commutative, Associative Laws }\\ 1+a^{?}& \text{ Complement Law }\\ 1& \text{ Identity Law }\end{array}$ $1$ a. $F(a,b)=\left(b^{?}+a\right)(a+b)$ $\begin{array}{l}(2\text{ points })\\ (2\text{ points })\\ (2\text{ points })\end{array}$ $\text{ b. }F(r,s,t)=r{s}^{?}+r(s+t{)}^{?}+s(s+t{)}^{?}$ $\text{ c. }F(x,y)=(x+y)+(y+xx)\left(x+y{y}^{?}\right)$ 2. For each of the following compact truth tables, use a Karnaugh map to obtain a simplified expression. a. $F(\underline{A},B,C)=?m(0,2,4,5,6,7)$ ( 2 points ) b. $F(\underline{\underline{A},\underline{B}},C)=?m(1,2,3,5,6,7)$ ( 2 points ) 3. (5 points) Design a simple combinational circuit which calculates the result of the function $F(n)=5n+3$, where $n$ is any 2-bit unsigned integer. In designing the circuit, draw a truth table, derive expressions for each of the outputs, and draw the circuit. You must show the process. You should not need a K-map. You must use Logisim to draw the circuit.