Venn diagram

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A Venn diagram is a visual way of representing the mathematical relationship between sets.

Two Set Example

The following diagram is a Venn diagram for sets $A$ and $B$ :

The red region contains all the elements that are in $A$ only. The blue region contains all the elements that are in $B$ only. The black region contains all the elements in both $A$ and $B$ which is called the intersection of $A$ and $B$ , denoted $A\cap B$ . The red, black, and blue regions together represent the elements that are in $A$ , $B$ , or both. This is called the union of $A$ and $B$ , denoted $A\cup B$ .

If we consider the region bounded by the rectangle to be the universal set, then the gray area is called the complement of $A\cup B$ -- that is, the things which are neither in $A$ nor in $B$ .

All of this information can be summarized in the following table:

Region (by color)	Description	Notation
Red	elements in $A$ only	$A - (A\cap B)$
Blue	elements in $B$ only	$B - (A\cap B)$
Black	elements in both $A$ and $B$	$(A\cap B)$
Gray	elements in neither $A$ nor $B$	$(A\cup B)^C$
		or $(A\cup B)'$
		or $\overline{(A\cup B)}$
		or $U - (A\cup B)$

Three Set Example

The following diagram is a Venn diagram for the sets $A, B$ and $C$ .

The following table describes the various regions in the diagram:

Region (by color)	Description	Notation
Blue	elements in $A$ only	$A - (A\cap B)-(C\cap A) + (A\cap B\cap C)$
Yellow	elements in $B$ only	$B - (A\cap B) - (B\cap C) + (A\cap B\cap C)$
Red	elements in $C$ only	$C - (B\cap C)-(C\cap A) + (A\cap B\cap C)$
Green	elements in both $A$ and $B$ but not $C$	$(A\cap B) - (A\cap B\cap C)$
Orange	elements in both $B$ and $C$ but not $A$	$(B\cap C) - (A\cap B\cap C)$
Purple	elements in $C$ and $A$ but not $B$	$(C\cap A) - (A\cap B\cap C)$
Black	elements in $A,B$ and $C$	$(A\cap B\cap C)$
Gray	elements in neither $A,B$ or $C$	$(A\cup B\cup C)^C$
		or $(A\cup B\cup C)'$
		or $\overline{(A\cup B\cup C)}$
		or $U - (A\cup B\cup C)$

Using Venn Diagrams

Venn diagrams are very useful for organizing data. In particular, the Principle of Inclusion-Exclusion can be explained for small cases nicely using them.

External links

A Survey of Venn Diagrams

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