What is a set?

A set is a collection of elements usually of the same kind. Each set can be denoted by a capital letter. A set may be defined by listing the members or describing the members. A set is said to be finite when all its members can be listed.

The above is an intuitive definition which seemed sufficient for several years but then contradictions such as Russel's paradox were pointed out, so a great deal of work had to be done to develop axioms that could define sets without allowing any paradoxes. These axioms require considerable mathematical sophistication to understand and often are the basis for a senior level college math

A set is a collection of items.

For example the set of furniture in a bedroom might be {bed, beside cupboard, dressing table, wardrobe, chair}; this is an example of a finite set as there are a finite number of elements in the set.

Another example is the counting numbers {1, 2, 3, ...}; this is an example of an infinite set as there is no end to the members of the set and it is impossible to list everyone of them.

A countable set is a set that can have each of its members put into relation with one of the counting numbers. Both the above example sets are countable.