Two of a Kind $[20$ points$]$

$($a$)$ Translate the Boolean Algebra expression $(x*Y{)}^{\text{'}}+(x^{\text{'}}+Y^{\text{'}}{)}^{\text{'}}*[(x*Y)+(x*Y^{\text{'}})+x^{\text{'}}]$ to Propositional

Logic. Use the variables $x$ and $y$ to represent the propositions $x=1$ and $Y=1,$ respectively. $[2$ points$]$

$($b$)$ Prove that your solution to $($a$)$ is a tautology using a chain of equivalences $($refering to axioms and theorems of boolean algebra$).[16$ points$]$

$($c$)$ Why do we know that the Boolean Algebra expression from part $($a$)$ will always evaluate to $1?$ Explain $(1-2$

sentences$).[2$ points$]$