Is it true that in any boolean algebra $x\oplus0 = x$?

Question

I was wondering about this question.

Logically, I would like to say that in every boolean algebra that is true.

Because: $x\oplus0=x·0'+x'·0 = x·0'$.

And in the binary boolean algebra that is correct as: $0'=1 \rightarrow x·0'=x·1=x$

But is that true for ANY of it?

I would love some insight from you, thank you!

@Shaun I figured because generally the members of the boolean algebra have to be a part of a certain group $B$, which can contain more than 2 members. — Eatay Mizrachi, Oct 14 '21 at 13:00
@William Yes, that is possibly true. I'm not studying in English, and we have the same name for both of these, I'll make an edit. I would appreciate any insight regarding the question. :) — Eatay Mizrachi, Oct 14 '21 at 13:25

score 3 · Answer 1 · answered Oct 14 '21 at 13:48

From the axioms of a boolean algebra, we have $$\lnot0=\lnot0\lor0=1$$ and $$x\land0=0\land x=0\land(0\lor x)=0,$$ so if $x\oplus y$ is defined as $(x\land\lnot y)\lor(\lnot x\land y)$, then we have $$x\oplus0=(x\land\lnot 0)\lor(\lnot x\land 0)=(x\land1)\lor0=x.$$

Is it true that in any boolean algebra $x\oplus0 = x$?

1 Answers1