Sets
| Site: | Newgate University Minna - Elearning Platform |
| Course: | Basic Mathematics |
| Book: | Sets |
| Printed by: | Guest user |
| Date: | Monday, 6 April 2026, 2:40 PM |
Description
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}
1. Subsets and Supersets
Consider two given sets A and B. If the set A consists of some or all elements of the
set B, then A is said to be a subset of B. We denote this by set notation ⊆. Thus A ⊆ B
means; A is a subset of B. if the set A is a subset of B with at least an element in B
not in A, then the set A is called a proper subset of B. Thus A ⊂ B means, A is a proper
subset of B. B is considered a superset of A.
The set notation for superset is ⊃. Thus B ⊃ A means B is a superset of A.
Example
If P = {2, 4, 6, 8,10}, Q = {4, 10}, R = {6, 8} and S = {2, 4, 6, 8, 10, 12}, then:
Q ⊂ P (ii) R ⊂ P (iii) S ⊃ P
In sets, the order in which the elements are written is irrelevant. For instance, {2, 4, 6,8} is the same as {2, 6, 8, 4}.
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
3. Complementary Set
Given that P is a subset of the universal set, the elements of the universal set which are not in the set P constitute a set which is called the complementary set of P. The
complement of P is denoted Pʹ
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}
5. The Intersection of Set
The intersection of two sets A and B; is the set which consists of elements that are in A as well as in B. The set notation for the operation of intersection is ∩. A ∩ B means; A intersection B.
In set theoretical A ∩ B = {X: X ∈ A and X ∈ B}.
6. Venn Diagram
Sets can be represented diagrammatically by closed figures. This method of set representation was developed by John Venn.
A Venn diagram is therefore a pictorial representation of sets. The operations of intersection, union and complementation of sets can easily be demonstrated by using Venn diagrams.