Are there no intersection points?

Question

It is given that $A= \{ x \in \mathbb{R}-\{0\}: sin(\frac{1}{x})=\frac{1}{x} \} $. To prove whether the given set A is compact or not

So if $x \in A \iff sin(\frac{1}{x})=\frac{1}{x}$

which means $1/y \in A \iff sin(y)=y$

and we know that the only fixed point of $sin$ is $0$ and as there exists no point, $x : 1/x=0$ and hence $A=\phi$ and hence $A$ is compact

ANOTHER ARGUMENT:

$sin(1/x)=1/x $ $\implies \exists $ a real number $1/x$ such that $sin(1/x)=1/x$ as we know the only fixed point of $sin$ is $0$ and hence $A$ is empty and hence compact

Are the both proofs correct? When I showed it to bing AI it said it looks wrong..

Answer 1

Yes, both proofs are correct. Bing AI is NOT a good way of verifying proofs, it does not understand maths. LLMs (Large Language Models) like Bing AI only understand the shape of sentences. It can generate sentences which look like sentences it has already seen. This can give it some information about truth. For example, it has likely seen a lot of sentences that say something like "the sky is blue", and few that say "the sky is red", so if you ask it what colour the sky is, it will correctly tell you blue. However, an incorrect maths proof looks a LOT like a correct maths proof, often just a single symbol off. If you prod it, you can get it to "prove" nonsense. For example, here is a dialogue I just had with ChatGPT:

Me: prove that 2 is a multiplicative identity

ChatGPT: To prove that 2 is a multiplicative identity, we need to show that for any real number (or any applicable number set) 'a', the following equation holds:

a * 2 = a

Let's proceed with the proof:

Let 'a' be any real number.

a * 2 = a

We know that multiplication is a commutative operation, which means the order of multiplication does not matter. Therefore, we can rewrite the left side of the equation as:

2 * a = a

Now, let's examine the right side of the equation. Since 'a' is a real number, we can use the distributive property of multiplication over addition:

2 * a = 1 * a + 1 * a

Now, let's simplify the right side:

2 * a = a + a

Since addition is also commutative, we can write this as:

2 * a = 2 * a

Now, subtract 2 * a from both sides of the equation:

2 * a - 2 * a = 2 * a - 2 * a

0 = 0

The equation is true, which means we have successfully shown that for any real number 'a', the equation holds true:

2 * a = a

This demonstrates that the number 2 acts as a multiplicative identity, as it leaves any real number 'a' unchanged when multiplied by 2.

You will note that it looks an awful lot like a maths proof, with only the small downside of being totally nonsense. If you try to get maths help from an LLM, it WILL give you incorrect answers, and worse, it will make them look like correct answers, because LLMs fundamentally do not understand mathematics. Ask a human instead.

This whole rant also applies to subjects other than mathematics. LLMs do not know anything about anything, and should never be relied upon.