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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?

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What happens if you add $x$ to both sides of the first equality ? –  Siméon Dec 6 '12 at 23:06
    
$x+x=x$ so both sides remain the same? –  Jason Byrne Dec 6 '12 at 23:14
    
Does '$+$' stands for 'xor' or for 'or' in your definition? –  Siméon Dec 6 '12 at 23:17
    
Oh sorry, '+' stands for 'or' –  Jason Byrne Dec 6 '12 at 23:18
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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 6 '12 at 23:21
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2 Answers

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

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Thanks joriki.. –  Jason Byrne Dec 7 '12 at 0:01
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Use absorption and distributivity: $$y=y(x+y)=y(x+z)=yx+yz=zx+yz=z(x+y)=z(x+z)=z.$$

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Thanks for the help. –  Jason Byrne Dec 7 '12 at 0:01
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