Venn diagram

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&cup;Y, contains all the elements in X and all those in Y.

Intersection
The intersection of X and Y, written X&cap;Y, contains all the elements that are common to both X and Y.

Set difference
The difference X minus Y, written X&minus;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&prime;, the complement of X, to be U&minus;X. X&prime; it contains everything in the universe apart from the elements of X.