Introduction to Set Theory
The concept of set is fundamental to mathematics and computer
science. Everything mathematical starts with sets. For example, relationships
between two objects are represented as a set of ordered pairs of objects, the
concept of ordered pair is defined using sets, natural numbers, which are the
basis of other numbers, are also defined using sets, the concept of function,
being a special type of relation, is based on sets, and graphs and digraphs
consisting of lines and points are described as an ordered pair of sets.
Though the concept of set is fundamental to mathematics, it is not defined rigorously here. Instead we rely on everyone's notion of "set" as a collection of objects or a container of objects. In that sense "set" is an undefined concept here. Similarly we say an object "belongs to " or "is a member of" a set without rigorously defining what it means. "An object(element) x belongs to a set A" is symbolically represented by "x A" . It is also assumed that sets have certain (obvious) properties usually asssociated with a collection of objects such as the union of sets exists, for any pair of sets there is a set that contains them etc.
This approach to set theory is called "naive set theory
" as opposed to more rigorous "axiomatic set theory". It was first
developed by the German mathematician Georg Cantor at the end of the 19th
century. For more on naive and axiomatic set theories