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. 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.