A while back, I came across this question. At the time, the question read:
What is $0 \neq 1$ in the context of Algebraic Axioms?
In this wiki link, the author included $0 \neq 1$ as the axiom that relates addition, and multiplication.
But, what I see the most is just the distributive law. What does $0 \neq 1$ mean? and how it is related to addition and multiplication?
I thought the question was very clear, so I answered it. I wrote:
Well, the statement $0 \ne 1$ means that $0$ is not the same thing as $1$. In other words, it means that the additive identity ($0$) is not the same as the multiplicative identity ($1$).
This statement is related to addition because it mentions the additive identity ($0$), and it's related to multiplication because it mentions the multiplicative identity ($1$).
I came back a while later, and I was surprised to see that my answer had received 3 downvotes and no upvotes. (I later deleted it.) So presumably there was something seriously wrong with it... but what?
I'm guessing that either I misunderstood the question, or I understood it correctly but my response didn't clarify anything.
Of the two possibilities, I think it's more likely that I misunderstood the question. In retrospect, "what does $0 \ne 1$ mean?" is a strange question, and I should have realized that someone who asks that question probably actually wants to know something else. So the only question at that point is, what is the intended question?
Matthew Daly's answer seems to have interpreted the question as, "Why is $0 \ne 1$ included as one of the ring axioms?" That answer got several upvotes and was accepted, so evidently that was the correct interpretation of the question. However, I'm very confused by that. I really don't see how that interpretation makes sense, given the question as written—to me, the questions "What does $0 \ne 1$ mean?" and "What was the motivation behind including $0 \ne 1$ in this list?" are completely different and unrelated questions.
So, in the future, when I come across questions similar to "What does $0 \ne 1$ mean," how can I determine the intended meaning, as Matthew apparently did for this question?
On the other hand, perhaps I did understand the question correctly, but my response failed to say anything that would help a student to understand anything. I admit that if someone doesn't know what $0 \ne 1$ means, then they're unlikely to have a strong understanding of the phrases "the additive identity" and "the multiplicative identity" either. Perhaps my answer would have been better if I'd given a more detailed explanation of just what the symbols $0$ and $1$ mean in the context of a ring, and of the resulting significance of the statement $0 \ne 1$.
So, I'm wondering:
- Did I misunderstand the question? If so, how could I have known what the correct interpretation was in this case?
- Did I understand the question correctly, but fail to provide any useful explanation?
- Or was the problem something else entirely?