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
3. Equality of Sets
Set A is equal to set B if they both have the same members, i.e if every element which belongs to A also belongs to B and if every element which belongs to B also belongs to A. We denote the equality of sets A and B by:
A = B
Example 5.1 Let A = {1, 2, 3, 4} and B = {3, 1, 4, 2}. Then A = B,
that is {1,2,3,4} = {3,1,4,2}, since each of the elements 1,2,3 and 4 of A belongs to B and each of the elements 3,1,4 and 2 of B belongs to A. Note therefore that a set does not change if its elements are rearranged.
Example 5.3 Let E={x | x²–3x = -2}, F={2,1} and G ={1,2,2, 1},
Then E= F= G