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I need to express a relationship between sets $A$ and $B$ such that $A\neq B$ and $A\cap B\neq\varnothing$. Is there a name for such a relation?

Can assume if needed that both are non empty.

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I just realised your title is not the same as the body of your question. The body simply says that at least one of $A$ or $B$ are non-empty. – Dan Rust Jan 21 '14 at 20:39
OK, actually, you can just say $A \neq B$. It follows automatically that one of them is nonempty, which is what the second part of your question is saying. – Dan Shved Jan 21 '14 at 20:42
But that answers the question in the body, not in the title. Maybe you should make them fit, otherwise it's not clear which one you mean. – Dan Shved Jan 21 '14 at 20:44
The answer to the question in the title is that A and B are 'not disjoint', here are some terms people use for this: see – George Tomlinson Jan 21 '14 at 20:55
The answer to the question in the body of the post is as Dan Shved says in his comment. – George Tomlinson Jan 21 '14 at 21:00

If I were writing something and needed to express this, I would write

Let $A$ and $B$ be distinct sets such that $A\cap B\neq\emptyset$.

or even just

Let $A$ and $B$ be distinct sets with non-empty intersection.

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Two sets $A$ and $B$ are disjoint if $A\cap B=\varnothing$. The most universally recognized vocabulary that you're looking for is "let $A$ and $B$ be distinct non-disjoint sets."

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hmm,but this does not cover the case where A==B, they are non-disjoint (as A intersection B is non empty == A == B) so, I think its not enough to say it. – Tzur Apr 2 '14 at 19:39

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