Give a Reason for Each Step in the Proof That

Essay

Give a reason for each step in the proof that x + xy = x is true in Boolean algebras. Your reasons should come from the following: associative laws for addition and multiplication, commutative laws for addition and multiplication, distributive law for multiplication over addition and distributive law for addition over multiplication, identity laws, unit property, zero property, and idempotent laws. $$\begin{array} { l } x + x y = x \cdot 1 + x y = x ( y + \bar { y } ) + x y = ( x y + x \bar { y } ) + x y = x y + ( x y + x \bar { y } ) = ( x y + x y ) + x \bar { y } = x y + x \bar { y } = \\x ( y + \bar { y } ) = x \cdot 1 = x\end{array}$$