Sets
In mathematics, we often en counter "sets", for example, real numbers from a set. Defining a set formally is a delicate matter, so we will use "naive" set theory, based on the intuitive properties of sets.
Definition :
A
set[1] is a collection of objects called elements. We use uppercase letters to label sets, and elements will usually be represented by lower case letters. When
is an element of a set
, we write
, otherwise, we write
if
contains no elements, it is the empty set, denoted
or
. Two sets are equal if they have exactly the same elements. In other words
.
Example :
The sets
and
are the same, because the ordering does not matter. The set
is also the same set as
, because we are not interested in repetitions.
Example :
The set
is implicit.
Definition :
The cardinal
of a set is number of distinct elements of
. If
is finite, the
is said to be finite. Otherwise,
is said to be infinite.
Example :
while
.
.
The set of primer numbers is infinite.