Let $A,B$ are two subsets of a Group $G$, and the product $AB=\{ab ;a\in A, b\in B\}$. Suppose that $AB\subset \emptyset$ then can I conclude that (either $A=\emptyset$ or $B=\emptyset$) or both of $A$ and $B$ is $\emptyset$?
About the product of two sets
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$\begingroup$
group-theory
elementary-set-theory
finite-groups
2 Answers
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Suppose that $A \ne \emptyset$ and $B \ne \emptyset$, then pick $a \in A$ and $b \in B$. We get $ab \in AB$. Therefore $AB$ is not a subset of $ \emptyset.$
Conclusion: if $AB\subset \emptyset$, then $A = \emptyset$ or $B = \emptyset$.
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The other answer gives you the correct implication. To complete this, it should be necessary to show that there are cases where $AB\subseteq \emptyset$ but (for instance) $B\neq\emptyset$. It seems that this is part of the OP's doubt.
So, take $A=\emptyset$ and $B$ any group. Let's read the definition of $AB=\{ab : a\in A, b\in B\}$ carefully: When $c\in AB$?
$c\in AB$ if and only if there exist $a\in A$ and $b\in B$ such that $c=ab$.
Since there exist no elements in $A$ whatsoever, there can be no such elements $c$ (regardless of $B$).
Set comprehensions (and every sort of definition) are tricky in borderline cases.