Sets
As mentioned in the introduction, a fundamental concept in all a branch of mathematics is that of set. Here is a definition “A set is any well-defined list, collection or class of objects”.
The objects in sets, as we shall see from examples, can be anything: But for clarity, we now list ten particular examples of sets:
Example 1.1 The numbers 0, 2, 4, 6, 8
Example 1.2 The solutions of the equation x²+ 2x+1 = 0
Example 1.3 The vowels of the alphabet: a, e, i, o, u
Example 1.4 The people living on earth
Example 1.5 The students Tom, Dick and Harry
Example1.6 The students who are absent from school
Example 1.7 The countries England, France and Denmark
Example 1.8 The capital cities of Nigeria
Example 1.9 The number 1, 3, 7, and 10
Example 1.10 The Rivers in Nigeria
Note that the sets in the odd numbered examples are defined, that is, presented, by actually listing its members; and the sets in the even numbered examples are defined by stating properties that is, rules, which decide whether or not a particular object is a member of the set
5. SUBSETS
If every element in a set A is also a member of a set B, then A is called subset of B.
More specifically, A is a subset of B if x∈A implies x∈B. We denote this relationship by writing; A⊂B, which can also be read “A is contained in B”.
Example 7.1
The set C = {1,3,5} is a subset of D = {5,4,3,2,1}, since each number 1, 3 and 5 belonging to C also belongs to D.
Example 7.2
The set E = {2,4,6} is a subset of F = {6,2,4}, since each number 2,4, and 6 belonging to E also belongs to F. Note, in particular, that E = F. In a similar manner it can be shown that every set is a subset of itself.
Example 7.3
Let G = {x | x is even}, i.e. G = {2,4,6}, and let F = {x | x is a positive power of 2}, i.e. let F = {2,4,8,16.....} Then F⊂G, i.e. F is contained in G.
With the above definition of a subset, we are able to restate the definition of the equality of two sets.
Two set A and B are equal, i.e. A = B, if an only if A⊂B and B⊂A. If A is a subset of B, then we can also write:
B⊃A
Which reads “B is a superset of A” or “B contains A”. Furthermore, we write:
A⊄B
if A is not a subset of B.
Conclusively, we state:
1. The null set ∅ is considered to be a subset of every set
2. If A is not a subset of B, that is, if A⊄B, then there is at least one element in A that is not a member of B.