ChatGPT and other large language models are not designed for calculation and will frequently be r/confidentlyincorrect in answering questions about mathematics; even if you subscribe to ChatGPT Plus and use its Wolfram|Alpha plugin, it's much better to go to Wolfram|Alpha directly. Even for more conceptual questions that don't require calculation, LLMs can lead you astray; they can also give you good ideas to investigate further, but you should never trust what an LLM tells you. To people reading this thread: DO NOT DOWNVOTE just because the OP mentioned or used an LLM to ask a mathematical question. I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns. More on reddit.com

It is perfectly fine to write $|A|>0$. ... empty set, rather than the cardinality of $A$. (The standard symbol in mathematics for "not equal" is $\neq$, rather than $!{=}$. You can make this symbol in $\LaTeX$ with the command \neq.) As mentioned in user21820's nice answer below, though, it is also very common to just write this in words ("$A$ is not empty" or "$A$ is nonempty") instead ... More on math.stackexchange.com

As far as I learned it, the empty set is a subset of all sets. So if both sets are empty than they are the same set. If bigset is nonEmpty and smallset is empty, then smallset is a subset of bigset. More on stackoverflow.com

The empty set is a concept, as all sets are. The empty set is not nothing. The empty set is a thing. It is a thing that contains nothing. As such, there are rules that govern proper reasoning so as to derive valid conclusions when your argument depends on the empty set. There is only one empty set. That is to say that it doesn't matter how a set comes to be empty, only that it is empty, the same as it would be if we were talking about some other subject matter. The empty set of people waiting at a particular bus stop, is the same object as the empty set of Chinese-Americans who have served as US President. There are all kinds of different logical systems that logicians have constructed for various reasons. Some are traditional, and some are what is known as "non-classical." (You can think of these non-classical systems as being like non-Eucildian geometry only as it applies to logic). They are constructed for various reasons, usually because the axioms have certain properties that the logician is interested in. The interesting thing to note on this point is that while the existence of a universal set (the infinite set containing all objects, often thought of as the opposite of the empty set) is controversial and there are many logical systems that specifically exclude the existence of the universal set, and even ones that exclude merely infinite sets (sets containing an infinite number of objects), the existence of the empty set is almost universally accepted. There are very few logical systems that deny the existence of the empty set. More on reddit.com