Mathematics-Online lexicon: Operations on Sets

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	Mathematics-Online lexicon:
	Operations on Sets

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

The following operations can be applied to sets

and

union:

$\displaystyle A\cup B=\{x:\ x\in A \ \lor \ x\in B\}\,,$
intersection:

$\displaystyle A\cap B=\{x:\ x\in A \ \land \ x\in B\}\,,$
difference, complement:

$\displaystyle A\setminus B=\{x:\ x\in A \ \land \ x \notin B\}\,,$
symmetric difference:

$\displaystyle A\Delta B= (A\setminus B) \cup (B \setminus A) = (A \cup B) \setminus (A\cap B)$

The so-called Venn-diagrams illustrate the set operations.

$\includegraphics[width=.3\moimagesize]{venndiagramm_A}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_B}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_AuB}$
		union: $A\cup B$
$\includegraphics[width=.3\moimagesize]{venndiagramm_AnB}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_AoB}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_AdB}$
intersection: $A \cap B$	difference: $A\setminus B$	symmetric difference: $A\Delta B$

If $B \subset A$ , some of the above diagrams are identical to one another:

$\includegraphics[width=.3\moimagesize]{venndiagramm_A2}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_B2}$	$\includegraphics[width=.3\moimagesize]{venndiagramm_AoB2}$
union: $A = A\cup B$	intersection: $B = A \cap B$	complement set : $A \setminus B = A \Delta B$

(Authors: Höllig/Hörner/Knesch/Abele)

Examples:

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automatically generated 6/11/2007