Sets
We are all familiar with the following collections:
1. A collection of books in a public library,
2. A collection of tools in a carpentry shop,
3. A collection of historical aircrafts in a museum.
Basically, any clearly defined collection of things, objects or numbers constitutes a set.
A collection of books in a public library is for instance, constitutes a set. Each member of a set is called an element of the set.
We shall use capital letters X, Y, Z etc. to denote sets, while small letters x, y, z, etc. will be used to denote the element of a set. This is purely conventional. When an element x, belongs to a set X, we write x ∈ X and say that x is a member or element of X. if x is not a member or an element of X, we write x ∉ X and we say that x does not belong to X.
A set is completely specified in the following ways:
1. By listing all the members of the set;
2. By describing the elements of the set;
3. By enclosing within braces, any general element with a clearly defined property, associated with the set. The symbol { } or ∅ denotes an empty set.
For instance:
A = {2, 3, 5, 7}, is the set consisting of prime numbers between 1 and 10, lists all the members of set A.
B is the set of all odd numbers between 1 and 10 describes the elements of the set B.
B = {3,5,7,9}
2. The Universal Set, the Unit Set and the Empty Set
The set which contains all the possible elements under consideration, is called the universal set.
The universal set is denoted