Boolean algebra- cancellation property of addition

[oneGirlArmy]

1. Prove that for all boolean algebras if x+y=x+z and x'+y = x'+z then y=z.


2. Homework Equations : x+x' = 1, xx'=0, basically we are allowed to use the usual boolean algebra properties.



3.Attempt: This the second part of a problem, in the first part we had to give and example of why x+y=x+z does not hold for all boolean algebras. Therefore this problem does not allow for cancellation of x on both sides to arrive at y=z. I have tried adding the LHS and RHS of both equations together to get x+x'+y+y = x+ x' +z+z which reduces to 1+y=1+z but I can't subtract the one from both sides in boolean algebra and I can't find a logical reason why they'd cancel. In fact, it seems the next logical reduction is to 1=1.

I also tried finding and expression for one equation in another and substituting to try and reduce one side to z and the other to x but I all I can come up with is complicated expressions that don't reduce very nicely and on my "best" attempts I've arrived at x+zy=x+yz

It seems like I am approaching the problem in the wrong way but I'm at a loss for another approach. Any suggestions?

 

[vela]

Did you try multiplying the two equations together? I didn't actually work it out by hand, but it looks like it should work.

 

[oneGirlArmy]

Yes, that does appear to work! Thanks so much that never would have occurred to me.