What is the difference between the cartesian product of two sets $(A \times B)$ and the intersection of those very same sets $(A \cap B)$? When I picture them on a graph I don't get the difference.
Thank you in advance for your answers.
J
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2$\begingroup$ What kind of graphs did you look at? Also, did you look at the definitions? $\endgroup$– WojowuOct 22, 2015 at 20:46
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$\begingroup$ Have you tried using a search engine for the definition of "Cartesian product" and "set intersection" - seriously, lazy. $\endgroup$– Alec TealOct 22, 2015 at 20:53
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1$\begingroup$ Who upvoted this (the spread is +1/-2) - show yourself! $\endgroup$– Alec TealOct 22, 2015 at 21:10
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1$\begingroup$ I guess I've misunderstood those concepts, but don't tell me I haven't looked it up. The definitions are kind of hard to grasp without thorough knowledge of mathematical notation, and the constructive answers I got definitely helped me understand where I was looking wrong. $\endgroup$– PxxOct 22, 2015 at 21:15
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1$\begingroup$ @Julien The accepted answer is nothing but a copy and paste of the definitions, so I hold the same position as Alec. $\endgroup$– YoTengoUnLCDOct 22, 2015 at 22:06
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3 Answers
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Given sets $A$ and $B$, the intersection $A \cap B$ is the set of elements which lie in both in $A$ and in $B$. Thus,
$A \cap B = \left\{x \bigm| x \in A \text{ and } x \in B \right\}$.
The Cartesian Product $A \times B$ is the set of all ordered pairs $(a,b)$ where $a \in A$ and $b \in B$. Thus,
$A \times B = \left\{(a,b) \bigm| a \in A \text{ and } b \in B\right\}$.
Henning's answer provides a concrete example of these definitions in action.
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$\begingroup$ You're welcome. Mathematical concepts can be tricky to grasp if it's your first pass at formal math; when in doubt, always fall back on the definitions. $\endgroup$ Oct 22, 2015 at 21:40
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Um, they have nothing in common, really. Consider, as a simple example $$ \{1,2\}\cap \{2,3\} = \{2\} \\ \{1,2\}\times\{2,3\} = \{\langle1,2\rangle,\langle1,3\rangle,\langle2,2\rangle,\langle2,3\rangle\}$$
You'd need some strange (but not impossible) contortions to find an example of $A$ and $B$ such that $A\cap B$ and $A\times B$ have even one element in common.
answered Oct 22, 2015 at 20:47
hmakholm left over Monicahmakholm left over Monica
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$\begingroup$ Thank you very much for your answer, that is very clear. $\endgroup$– PxxOct 22, 2015 at 21:15
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Cartesian product between two sets $A$ and $B$ is the set of couples $A\times B:=\{(a,b);a\in A \text{ and }b\in B\}$, whereas $A\cap B$ is the set $A\cap B:=\{c;c\in A\text{ and }c\in B\}$.
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$\begingroup$ My confusion was in the difference between (a,b) and (c). Thank you very much for your answer. $\endgroup$– PxxOct 22, 2015 at 21:16
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