Venn Diagrams

The Venn diagram, is a convenient way to illustrate definitions within the algebra of sets. Consider a Universal set with two subsets A and B. We may represent this as a rectange containing the universal set, with circles containing the elements of A and B.
Venn diagram

The complement of a set A is everything that is not in A; it is represented by the magenta region in the Venn diagram below (hence the set A is represented by the white region).

A complement

The union of A and B is everything which is in either A or B, as represented by the magenta shaded region in the following venn diagram.


The intersection of two sets is that which is in both sets, as represented by the magenta shaded region in the following Venn diagram.


The four regions into which a Venn diagram with two circles divides the universal set can be identified as intersections of the two subsets and their complements as labelled in the following Venn diagram.

Labelled regions

Two sets are mutually exclusive (also called disjoint) if they do not have any elements in common; they need not together comprise the universal set. The following Venn diagram represents mutually exclusive (disjoint) sets.

Mutually exclusive sets

If the union of two mutually exclusive sets is the universal set they are called complementary. The intersection of two complementary sets is the null set, and the union is the universal set, as the following Venn diagram suggests.

Complementary sets

Venn diagrams can also help motivate some definitions and laws in probability. From the basic two circle Venn diagram above, it is easy to see that P(AUB) = P(A) + P(B) - P(AB) because the intersection (AB) is included in both A and B. The definition of conditional probability P(A|B) (read probability of A conditioned on B) may be motivated by the following Venn diagram. The universal set is replaced by the set B which is being conditioned on, hence one is only interested in that portion of A which is in B, and its probability relative to the set B.


return to index