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
7. Comparability
Two sets A and B are said to be comparable if:
A⊂B or B⊂A;
That is, if one of the sets is a subset of the other set. Moreover, two sets A and B are said to be not comparable if:
A⊄B and B⊄A
Note that if A is not comparable to B then there is an element in A.which is not in B and ... also, there is an element in B which is not in A.
Example 8.1: Let A = {a,b} and B = {a,b,c}. The A is comparable to B, since
A is a subset of B.
Example 8.2: Let R – {a,b} and S = {b,c,d}. Then R and S are not comparable, since a∈R and a∉S and c∉R
In mathematics, many statements can be proven to be true by the use of previous assumptions and definitions. In fact, the essence of mathematics consists of theorems and their proofs. We now proof our first
Theorem 1.1: If A is a subset of B and B is a subset of C then A is a subset of
C, that is,
A⊂B and B⊂C implies A⊂B
Proof: (Notice that we must show that any element in A is also an element in C). Let x be an element of A, that is, let x∈A. Since A is a subset of B, x also belongs to B, that is, x∈B. But by hypothesis, B⊂C; hence every element of B, which includes x, is a number of C. We have shown that x∈A implies x∈C.
Accordingly, by definition, A⊂C.