Proving at boolean algebra

Question

Must prove that $$(X+Y )=X+(X.Y')$$

i tried a lot of ways, using logic things and expanding this things, but cant reach the Y. $$(X+Y )=(X+X).(X+Y')$$

Whats the possible prove to this?

Answer 1

It should go like this: \begin{align*} X + Y &= (X + Y) \cdot 1 = (X + Y) \cdot (X + X') = XX + XX' + YX + YX' \\ &= X + 0 + XY + X'Y = X + XY + X'Y = X(1+Y) + X'Y = X \cdot 1 + X'Y \\ &= X + X'Y. \end{align*}