Boolean algebra proof - point me in the right direction?

Question

I wish to formulate a proof that if $x+y = x+z$ and $xy$ = $xz$ then $y=z$. I'm just beginning my study of Boolean algebra, but is $y=z$ not self evident from the stated equations?

What happens if you add $x$ to both sides of the first equality ? — Siméon, Dec 06 '12 at 23:06
Ok, my mistake then. From $x+y=x+z$ you can derive $(\neg x)y = (\neg x)z$ and then compute $(\neg x + x)y = \dots$. — Siméon, Dec 06 '12 at 23:21

score 2 · Answer 1 · answered Dec 06 '12 at 23:24

2

$x$ is either true or false. In either case you can derive $y=z$ from one of the equations.

answered Dec 06 '12 at 23:24

joriki

238,052

score 0 · Answer 2 · answered Dec 06 '12 at 23:34

0

Use absorption and distributivity: $$y=y(x+y)=y(x+z)=yx+yz=zx+yz=z(x+y)=z(x+z)=z.$$

answered Dec 06 '12 at 23:34

Berci

90,745

Boolean algebra proof - point me in the right direction?

2 Answers2