Can I factor out or statements on the other side of an equation in boolean?

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I have this boolean equation:

X'Y'+XY+X'Y=X'+Y

I want to prove it.

Now I was wondering if I can rearrange this equation, if I could, so I can factor out the other side; tell me if this is allowed. I haven't seen anything to say I could in my textbook:

X'Y'+XY+X'Y

X'Y'+X'Y+XY see now I move the X'Y to the left

X'(Y+Y')+XY
X'+X'Y+XY
X'+Y(X'+X)
X'+Y

Am I doing it right? I've been trying this equation in other ways and haven't been able to prove it otherwise.

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1 Answer

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Almost all of your rearrangements are correct, except it is not clear how you get from $X'(Y+Y')+XY$ to $X'+X'Y+XY$. I would write your argument like this: $$\begin{split}X'Y'+XY+X'Y&=X'Y'+X'Y+XY\\ &=X'(Y'+Y)+XY\\ &=X'(1)+XY\\ &=X'+XY\\ &=(X'+X)(X'+Y)\\ &=(1)(X'+Y)\\ &=X'+Y.\end{split}$$

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