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}
4. The Union of Sets
The union of set A and B, is the set which consists of elements that are either in A or B or both. The set notation for the operation of union is ⋃. Thus, A union B is written
A ⋃ B.
In set theoretical notation,
A ⋃ B = {X : X ∈ A, or X ∈ B or X ∈ both A and B}.
Example
Given that G= {h, e, a, p}, H = {l, a, k, e} then,
G ⋃ H = {h, e, a, p, l, k}
If X= {2, 4, 6, 8}, Y = {2, 5, 9, 11, 12}
Then X ⋃ Y = {2, 4, 5, 6, 8, 9, 11, 12}