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Universal Set: Definition

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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):
universal set


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

Stephanie Glen. "Universal Set: Definition" From Calculus for the rest of us!

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