Let $A = \{1,2\}$ and $B = \{x,y\}$ Does this notation $(A,B)$ mean $\{(1,x), (1,y), (2,x), (2,y)\}$? That is, does it mean the Cartesian product $A\times B$?
About Sets and Ordered Pairs
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elementary-set-theory
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3I would interpret it as an ordered pair of sets. You would need to provide more context to comfortably disambiguate. – 2017-02-11
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0It *probably* means that incorrectly. But *coorectly* it means something entirely different. Let $K = P(\mathbb R)$ = the set of all subsets of $\mathbb R$. Let $M =$ the set of all sets of variables. Then $A \in K$ and $B \in M$ and $(A,B)$ is an ordered pair in $K \times M$. – 2017-02-11
2 Answers
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The notation $(A,B)$ is a bit ambiguous but it would not normally be interpreted as the cartesian product but rather as $(\{1,2\}, \{x,y\})$. That is the ordered pair of sets $(A,B)$.
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No, I wouldn't says so, it's just a pair $(\{1,2\},\{x,y\})$ ,I would say. analogous to formulations like that a topological space is a pair $(X, \mathcal{T})$, where $\mathcal{T} \subset \mathscr{P}(X)$ obeys axioms.