# Universal Set: Definition

Share on

A universal set, denoted by U, has all elements of interest. Elements of a set are a collection of items, so you can also think of the universal set as the overall collection of items you’re interested in: If you’re interested in studying freshmen students, then “all freshmen” is you’re universal set; if you’re studying the ethnic make up of political parties, then U = all political parties. Anything you think of can be defined within a universal sets, from frogs to footballs, from numbers to electrons.

All other sets are subsets of the universal set. For example, a list of the 52 U.S. states is a universal set, with subsets of Eastern states, Western states, states that begin with the letter A, and so on.

Defining U is helpful for establishing a frame of reference for set problems [1]. The rule for a set is that each member of U has to be clearly in the set, or not in the set. For example, you probably would not have U as a list of nice chocolate manufacturers, as there’s some ambiguity as to what “nice” is.

## Does the Universal Set Contain Everything?

The idea of the universal set has been around from before the 20th century, when mathematicians and philosophers first imagined a collection of all possible entities [2]. However, some set theories do not allow U to contain everything; Cantor and Bertrand Russell proved that U cannot contain everything as it leads to paradoxes and inconsistencies. Other set theories (such as Zermelo–Fraenkel set theory) simply do not include U at all.

## Venn Diagram of the Universal Set

The following Venn diagram shows the universal set with a subset A, a subset of interest (left) and with A as a subset of B, and both are subsets of U (right):

## References

[1] Wooland. Part 1 Module. Set Mathematics Sets, Elements, Subsets. Article posted on Florida State University website.