Simplify, using algebraic manipulations, the following Boolean expressions to a minimum num$-$ ber of terms and factors.

$($a$)$ XYZ$+$XY $+$XYZ

$($b$)$ XYZ$+$XZ

Find the complement of the following expression: $($a$)$ XY $+$XY

Propose and solve your own logic simplification problem.

Simplify the following expressions using K$$Map

$($a$)$ XY $+$YZ$+$XYZ $$

Simplify the following expressions by means of a four$-$ variable K$$Map

$($a$)$ AD$+$BD$+$BC$+$ABD $($b$)$ ABC$+$CD$+$BCD$+$BC