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
8. Sets of Sets
It sometimes will happen that the object of a set are sets themselves; for example, the set of all subsets of A. In order to avoid saying “set of sets”, it is common practice to say “family of sets” or “class of sets”. Under the circumstances, and in order to avoid confusion, we sometimes will let script
letters A, B,..........
Denote families, or classes, of sets since capital letters already denote their elements.
Example 9.1: In geometry we usually say “a family of lines” or “a family of curves” since lines and curves are themselves sets of points.
Example 9.2: The set {{2,3}, {2}, {5,6}} is a family of sets. Its members are the sets {2,3}, {2} and {5,6}.
Theoretically, it is possible that a set has some members, which are sets themselves and some members which are not sets, although in any application of the theory of sets this case arises infrequently
Example 9.3: Let A = {2, {1,3}, 4, {2,5}}. Then A is not a family of sets; here some elements of A are sets and some are not.