# 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.

J

• What kind of graphs did you look at? Also, did you look at the definitions? – Wojowu Oct 22 '15 at 20:46
• Have you tried using a search engine for the definition of "Cartesian product" and "set intersection" - seriously, lazy. – Alec Teal Oct 22 '15 at 20:53
• Who upvoted this (the spread is +1/-2) - show yourself! – Alec Teal Oct 22 '15 at 21:10
• 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. – Jxx Oct 22 '15 at 21:15
• @Julien The accepted answer is nothing but a copy and paste of the definitions, so I hold the same position as Alec. – YoTengoUnLCD Oct 22 '15 at 22:06

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.

• Thank you very much, I understood it now. :) – Jxx Oct 22 '15 at 21:16
• 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. – Gaussian0617 Oct 22 '15 at 21:40

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.

• Thank you very much for your answer, that is very clear. – Jxx Oct 22 '15 at 21:15

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 $A\cap B:=\{c;c\in A\text{ and }c\in B\}$.

• My confusion was in the difference between (a,b) and (c). Thank you very much for your answer. – Jxx Oct 22 '15 at 21:16
• You're welcome ! – Balloon Oct 22 '15 at 21:29