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
14. SUMMARY
A summary of the basic concept of set theory is as follows:
A set is any well-defined list, collection, or class of objects.
Given a set A with elements 1,3,5,7 the tabular form of representing this set is A = {1, 3, 5, 7}.
The set-builder form of the same set is A = {x| x = 2n + 1,0 ≤n≤3}
Given the set N = {2,4,6,8,....} then N is said to be infinite, since the counting process of its elements will never come to an end, otherwise it is finite
Two sets of A and B are said to be equal if they both have the same elements, written A = B
The null set,∅ , contains no elements and is a subset of every set
The set A is a subset of another set B, written A⊂B, if every element of A is also an element of B, i.e. for every x∈A then x∈B
If B⊂A and B≠A, then B is a proper subset of A
Two sets A and B are comparable if A⊂B and B⊂A
The power set 2² of any set S is the family of all the subsets of S
Two sets A and B are said to be disjoint if they do not have any element in common, i.e their intersection is a null set.