Some of the common symbols are a bit tricky to type on ascii keyboards. I use vim's digraph feature to enter them.

A set is a group of objects represented as a unit. Sets may contain any type of object including numbers, symbols, and even other sets. The objects in a set are called it's elements or members. Set may be described formally in several ways.

definition

One was is by listing a set's elements inside braces. Thus the set: S = {7,21,57} contains 7, 21, and 57. The symbols ∈ and ∉ denote set membership and non-membership. We write 7 ∈ {7,21,57} and 8 ∉ {7,21,57}. For two sets A and B, we say that A is a subset of B, written A ⊆ B, if every member of A is also a member of B. We say that A is a proper subset of B, written A ⊊ B, if A is a subset of B and is not equal to B.

order

The order of describing a set doesn't matter, nor does repetition of its members. We get the same set S by writing {57,7,7,7,21}. If we do want to take the number of occurrences of members into account, we call the group a multiset instead of a set. Thus {7} and {7,7} are different as multisets, but identical as sets.

infinite and finite

An infinite set contains infinitely many elements. We cannot write a list of all the elements of an infinite set, so we sometimes use the ... notation to mean "continue the sequence forever". Thus we the set of natural numbers ℕ as {1,2,3,...}. The set of integers ℤ as {...,-2,-1,0,1,2,...}. The set with zero members is sometimes called the empty set and is written ∅. A set with one member is sometimes called a singleton set, and a set with two members is called an unordered pair.

rules

When we want to describe a set containing elements according to some rule, we write {n|rule about n}. Thus { n|n = m² for some m ∈ ℕ } means the set of perfect squares.

unions

If we have two sets A and B, the union of A and B, written A ∪ B, is the set we get be combining all the elements in A and B into a single set. The intersection of A and B, written A ∩ B, is the set of elements that are in both A and B. The complement of A, written Ā, is the set of all elements under consideration that are not in A.