set

Sets and their relationships are often represented by Venn diagrams. The sets are drawn as circles - the area of overlap between the circles shows elements that are common to each set, and thus represent a third set. (Top) A Venn diagram of two intersecting sets and (bottom) a Venn diagram showing the set of whole numbers from 1 to 20 and the subsets P and O of prime and odd numbers, respectively. The intersection of P and O contains all the prime numbers that are also odd.

In mathematics, any collection of defined things (elements), provided the elements are distinct and that there is a rule to decide whether an element is a member of a set. It is usually denoted by a capital letter and indicated by curly brackets {}.

For example, L may represent the set that consists of all the letters of the alphabet. The symbol ∈ stands for ‘is a member of’; thus p ∈ L means that p belongs to the set consisting of all letters, and 4 ∉ L means that 4 does not belong to the set consisting of all letters.

There are various types of sets. A finite set has a limited number of members, such as the letters of the alphabet; an infinite set has an unlimited number of members, such as all whole numbers; an empty or null set has no members, such as the number of people who have swum across the Atlantic Ocean, written as {} or ø; a single-element set has only one member, such as days of the week beginning with M, written as {Monday}. Equal sets have the same members; for example, if W = {days of the week} and S = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}, it can be said that W = S. Sets with the same number of members are equivalent sets. Sets with some members in common are intersecting sets; for example, if R = {red playing cards} and F = {face cards}, then R and F share the members that are red face cards. Sets with no members in common are disjoint sets. Sets contained within others are subsets; for example, V = {vowels} is a subset of L = {letters of the alphabet}.

Sets and their interrelationships are often illustrated by a Venn diagram.