Set

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

Generally a set is indicated by a capital letter. The elements of the set are indicated by lowercase letters.

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".

Examples

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

$$ 2.5 \in Q, R $$

$$ 2.5 \notin N, Z $$