A set is a collection of objects called elements. The order of the elements in the set is irrelevant.

A set is a grouping of objects determined by an objective criterion that uniquely establishes the membership or non-membership of any object in the set. The objects that make up a set are called elements.

Generally a set is indicated by a capital letter (A, B, C, ...). The elements of the set are indicated by lowercase letters (a,b,c,...).

Set Types

Based on the number of elements, a set can be defined as follows:

  • Finite set. A finite set is composed of a definite number of elements. For example, the set of months in a year and the set of days in a month.
  • Infinite set. An infinite set is composed of an infinite number of elements. For example, the set of natural numbers and the set of real numbers.
  • Empty set. An empty set is a set that has no elements. The empty set is represented by the symbol Ø. An example of an empty set is the set of squares with three sides.
  • Universal set. The universal set is the set that contains all the elements of the other sets. The universal set is unique and is generally indicated by the uppercase letter U. All sets are subsets of the universal set.

Set Membership Criterion

A set is defined by a membership criterion that unambiguously determines whether an object belongs or does not belong to a set. Each element of the set appears only once, and unlike vectors and ordered sets, it does not have a specified order. The membership criterion for a set is indicated by the symbol ∈.

x ∈ Y

The membership of an element x in set Y is written as "x ∈ Y" and read as "x belongs to Y". Similarly, it is possible to indicate the criterion of non-membership in the set by the barred symbol ∉. The non-membership of element x in set Z is written as "x ∉ Z" and read as "x does not belong to Z".

Let A be a set and let p, q be objects

  • when p is an element of A we write p∈A.
  • when q is not an element of A we write q∉A.

The symbol ∈ indicates membership. It may be translated as "be in", "are in", "is in", "in", "belong to" according to context.

Thus, "let n ∈ N" may be read as "let n be in N".

$$ n \in N $$

"∀ n,q ∈ N" may be read as "for any n and q in N"

$$ \forall \ n,q \in N $$

A set is well defined if it is always possible to determine whether each object does or does not belong to the set.

Sometimes a set is given in tabular form by indicating its elements between braces

$$ A = \{ \ a \ , \ e \ , \ i \ , \ o \ , \ u \ \} $$

$$ B = \{ 2 \ , \ 4 \ , \ 6 \ ... \} $$

Other times it is represented by the conditions of the set between braces.

$$ C = \{ x \ : \ x \in N \ , \ x \ is \ even \} $$

The set C consists of all objects x satisfying the conditions "x is a natural number" and "x is even".


N is the set of natural numbers

The number 1 ∈ N since 1 is a natural number

The number -1 ∉ N since -1 is not a natural number

There are other numeric sets

  • Z is the set of integers
  • Q is the set of rational numbers
  • R is the set of all real numbers
  • C is the set of complex numbers

An object can be an element of multiple sets

For example, the number -1 is a integer, a rational number and a real number but not an natural number. $$ -1 \in Z, Q, R $$ $$ -1 \notin N $$ For example, the number 2.5 is a rational number and real number but not an integer or natural number $$ 2.5 \in Q, R $$ $$ 2.5 \notin N, Z $$

The Representation of Sets

A set can be represented through graphic representation, tabular representation (representation by listing), or characteristic property representation.

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