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

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.

## 1 Answer

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: $$\begin{split}X'Y'+XY+X'Y&=X'Y'+X'Y+XY\\ &=X'(Y'+Y)+XY\\ &=X'(1)+XY\\ &=X'+XY\\ &=(X'+X)(X'+Y)\\ &=(1)(X'+Y)\\ &=X'+Y.\end{split}$$

• Are there any specific limitations to factoring in boolean? Can I factor out of 3+ statements or any part of the expression? – munchschair Feb 11 '14 at 3:39
• What specifically do you mean by "factoring"? Because I would call what you have done "simplifying". You can certainly simplify boolean expressions that have 3+ terms. – Dave Wilding Feb 11 '14 at 9:15
• To help me understand: could you tell me what you see in $X'Y'+XY+X'Y$ that makes you want to get $X'+Y$ rather than $X+Y$? – Dave Wilding Feb 11 '14 at 9:17
• It's a proof that you can get to that. Factoring as in pulling alike terms out of the multiples. – munchschair Feb 11 '14 at 12:14
• OK, I see. When you are working in boolean you can factor in two different ways: $X_1Y+X_2Y+\dotsb +X_nY=(X_1+\dotsb +X_n)Y$ and $(X_1X_2\dotsm X_n)+Y=(X_1+Y)(X_2+Y)\dotsm(X_n+Y)$ for any number of $X_1,X_2,\dotsc,X_n$. – Dave Wilding Feb 11 '14 at 12:26