Venn Diagrams

Sets and the relationships between them can be pictured using simple diagrams

introduced by the British logician John Venn, called Venn diagrams. They are

an extremely useful visual tool for understanding set operations before computing

them algebraically.

Convention. The universal set $U$ is represented by a rectangle. Every other set

under discussion is represented by a circle (or closed curve) drawn inside that

rectangle, since every such set is a subset of $U$. If a set $A$ is a subset of

another set $B$, the circle for $A$ is drawn entirely inside the circle for $B$. If

two sets $A$ and $B$ have some elements in common but neither is a subset of the

other, their circles are drawn overlapping, so that the overlapping (lens-shaped)

region represents the elements common to both.

Reading a Venn diagram. For two sets $A$ and $B$ drawn as overlapping circles

inside a rectangle $U$:

• The region where the two circles overlap represents $A \cap B$ (elements in both $A$ and $B$).
• The part of circle $A$ outside the overlap represents $A - B$ (elements in $A$ but not in $B$).
• The part of circle $B$ outside the overlap represents $B - A$ (elements in $B$ but not in $A$).
• The whole shaded area covered by either circle (overlap included, counted once) represents $A \cup B$.
• The region inside the rectangle but outside both circles represents the complement $(A \cup B)'$ — elements of $U$ belonging to neither set.

Worked illustration. Take $U = \{1, 2, \dots, 10\}$, $A = \{1, 2, 3, 4\}$,

$B = \{3, 4, 5, 6\}$. Drawing two overlapping circles for $A$ and $B$ inside a

rectangle for $U$: the elements $1, 2$ are placed in the part of circle $A$ that does

not overlap $B$ (since they belong only to $A$); the elements $5, 6$ are placed in the

part of circle $B$ that does not overlap $A$; the elements $3, 4$ — which belong to

both sets — are placed in the overlapping lens; and the remaining elements

$7, 8, 9, 10$, belonging to neither $A$ nor $B$, are placed inside the rectangle but …