mathematics and logic

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- Newcastle University - Sets and Venn Diagrams
- University of California, Davis - Department of Mathematics - Sets and Functions
- Massachusetts Institute of Technology - Department of Mathematics - Sets, Numbers, and Logic
- Mathematics LibreTexts Library - Sets
- University College London - Sets and functions
- Texas A and M university Technology Services - Sets and Probability
- Maths Is Fun - Introduction to Set
- Whitman College - Logic and Sets
- Indiana University - Set Theory
- Story of Mathematics - Describing Sets – Methods and Examples
- Carnegie Mellon University - Mathematical Sciences - Sets

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External Websites

- Newcastle University - Sets and Venn Diagrams
- University of California, Davis - Department of Mathematics - Sets and Functions
- Massachusetts Institute of Technology - Department of Mathematics - Sets, Numbers, and Logic
- Mathematics LibreTexts Library - Sets
- University College London - Sets and functions
- Texas A and M university Technology Services - Sets and Probability
- Maths Is Fun - Introduction to Set
- Whitman College - Logic and Sets
- Indiana University - Set Theory
- Story of Mathematics - Describing Sets – Methods and Examples
- Carnegie Mellon University - Mathematical Sciences - Sets

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**set**, in mathematics and logic, any collection of objects (elements), which may be mathematical (e.g., numbers and functions) or not. A set is commonly represented as a list of all its members enclosed in braces. The intuitive idea of a set is probably even older than that of number. Members of a herd of animals, for example, could be matched with stones in a sack without members of either set actually being counted. The notion extends into the infinite. For example, the set of integers from 1 to 100 is finite, whereas the set of all integers is infinite. Because an infinite set cannot be listed, it is usually represented by a formula that generates its elements when applied to the elements of the set of counting numbers. Thus, {2*x* | *x* = 1, 2, 3,…} represents the set of positive even numbers (the vertical bar means “such that”).

To indicate that an object *x* is a member of a set *A*, one writes *x* ∊ *A*, while *x* ∉ *A* indicates that *x* is not a member of *A*. A set with no members is called an empty, or null, set and is denoted ∅. A set *A* is called a subset of a set *B* (symbolized by *A* ⊆ *B*) if all the members of *A* are also members of *B*. For example, any set is a subset of itself, and Ø is a subset of any set. If both *A* ⊆ *B* and *B* ⊆ *A*, then *A* and *B* have exactly the same members. Part of the set concept is that in this case *A* = *B*; that is, *A* and *B* are the same set.

The symbol ∪ is employed to denote the union of two sets. Thus, the set *A* ∪ *B*—read “*A* union *B*” or “the union of *A* and *B*”—is defined as the set that consists of all elements belonging to either set *A* or set *B* or both. For example, if the set *A* is given by {1, 2, 3, 4, 5} and the set *B* is given by {1, 3, 5, 7, 9}, the set *A* ∪ *B* is {1, 2, 3, 4, 5, 7, 9}.

The intersection operation is denoted by the symbol ∩. The set *A* ∩ *B*—read “*A* intersection *B*” or “the intersection of *A* and *B*”—is defined as the set composed of all elements that belong to both *A* and *B*. Thus, the intersection of the two sets in the previous example is the set {1, 3, 5}. For more information about sets and their use in mathematics, *see* set theory.

The Editors of Encyclopaedia BritannicaThis article was most recently revised and updated by Erik Gregersen.