Due date: $\frac{10}{6}11$:$50$pm

Question $1$: Prove this equation $(1$ point$)$

$(x_1+x_2+x_3)*(x_1+x_2+{\stackrel{}{x}}_3)=x_1+x_2$

Question $2$: $(2$ points$)$

Consider the logic expressions

$f=x_1{\stackrel{}{x}}_2{\stackrel{}{x}}_5+{\stackrel{}{x}}_1{\stackrel{}{x}}_2{\stackrel{}{x}}_4{\stackrel{}{x}}_5+x_1{x}_2{x}_4{x}_5+{\stackrel{}{x}}_1{\stackrel{}{x}}_2{x}_3{\stackrel{}{x}}_4+x_1{\stackrel{}{x}}_2{x}_3{x}_5+{\stackrel{}{x}}_2{\stackrel{}{x}}_3{x}_4{\stackrel{}{x}}_5+x_1{x}_2{x}_3{x}_4{\stackrel{}{x}}_5$

$g={\stackrel{}{x}}_2{x}_3{\stackrel{}{x}}_4+{\stackrel{}{x}}_2{\stackrel{}{x}}_3{\stackrel{}{x}}_4{\stackrel{}{x}}_5+x_1{x}_3{x}_4{\stackrel{}{x}}_5+x_1{\stackrel{}{x}}_2{x}_4{\stackrel{}{x}}_5+x_1{x}_3{x}_4{x}_5+{\stackrel{}{x}}_1{\stackrel{}{x}}_2{\stackrel{}{x}}_3{\stackrel{}{x}}_5+x_1{x}_2{\stackrel{}{x}}_3{x}_4{x}_5$

Prove or disprove that $f=g.$

Question $3$: Use your knowledge of DeMorgan's laws and Boolean Algebra to simplify the following circuit. A and $B$ are inputs, and OUT is the output. $(2$ points$)$