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
2. 2. Finite & Infinite Sets
Sets can be finite or infinite. Intuitively, a set is finite if it consists of a specific number of different elements, i.e. if in counting the different members of the
set the counting process can come to an end. Otherwise a set is infinite. Lets look at some examples.
Example 4:1: Let M be the set of the days of the week. The M is finite
Example 4:2: Let N = {0,2,4,6,8........}. Then N is infinite
Example 4:3: Let P = {x< x is a river on the earth}. Although it maybe difficult to count the number of rivers in the world, P is still a finite set.
Exercise 1.3: Which sets are finite?
1. The months of the year
2. {1, 2, 3, ......... 99, 100}
3. The people living on the earth
4. {x | x is even}
5. {1, 2, 3,........}
Solution:
The first three sets are finite. Although physically it might be impossible to count the number of people on the earth, the set is still finite. The last two sets are infinite. If we ever try to count the even numbers, we would never come to the end.