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
$$ 2.5 \in Q, R $$
$$ 2.5 \notin N, Z $$