Venn diagram

(PD) Image: John R. Brews
Venn diagrams; set X is the blue circle (left) and its interior, set Y is the red circle (right) and its interior. The rectangle represents the universal set.

A Venn diagram is an arrangement of intersecting circles used to visually represent logical concepts and propositions.

Examples

Suppose X and Y are sets, for example, collections of some description. Various operations allow us to build new sets from them, and these definitions are illustrated using the three Venn diagrams in the figure.

Union

The union of X and Y, written X∪Y, contains all the elements in X and all those in Y.

Intersection

The intersection of X and Y, written X∩Y, contains all the elements that are common to both X and Y.

Set difference

The difference X minus Y, written X−Y or X\Y, contains all those elements in X that are not also in Y.

Complement and universal set

The universal set (if it exists), usually denoted U, is a set of which everything under discussion is a member. In pure set theory, normally sets are the only objects considered. In that case, U would be the set of all sets. However, one may also consider sets that are collections of numbers, or colors, or books, for example; see Set (mathematics).

In the presence of a universal set we can define X′, the complement of X, to be U−X. X′ it contains everything in the universe apart from the elements of X.