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
10. Power Set
The family of all the subsets of any set S is called the power set of S.
We denote the power set of S by: 2S
Example 11.1: Let M = {a,b} Then 2
M = {{a, b}, {a}, {b}, ∅}
Example 11.2: Let T = {4,7,8} then 2
T = {T, {4,7}, {4,8}, {7,8}, {4}, {7},
{8}, ∅}
If a set S is finite, say S has n elements, then the power set of S can be shown to have 2n elements. This is one reason why the class of subsets of S is called
the power set of S and is denoted by 2 S.