What is the difference between cartesian product and intersection? 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
 A: 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.
A: 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.
A: 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 $A\cap B:=\{c;c\in A\text{ and }c\in B\}$.
