There’s one more technique that we’ll look at here. It is known as mathematical induction. A less formal term might be proof by mathematical chain reaction. Using this scheme, it is possible to prove things about all the elements in an infi- nite set, using only a finite number of steps.
Imagine an infinite set S, consisting of elements called a0, a1, a2, a3, a4, and so on, like this:
S= {a0, a1, a2, a3, a4, . . .}
Suppose that we want to prove that a proposition P is true about all the elements of S. We can’t prove P for each element of Sone by one, or even for large batches of elements, because the list goes on forever. But suppose we can prove that P is true for a0, the first element in S. Also suppose we can prove that if P is true for some unspecified element anin set S(where nis a natural number), then P is true for the next element a(n+1)in set S. By doing these two things, we create a “chain reaction of truths.” We know P is true for the first element, and this proves that P is true for the second; that in turn proves P for the third; and so it goes on for- ever, like an infinitely long line of dominoes knocking each other down.
PROBLEM 3-6
Show that for any two distinct rational numbers, there is a third rational number whose value lies between them. Don’t use reductio ad absur- dum, and don’t try to use mathematical induction. It is all right, how- ever, to take all the general rules of arithmetic (sums, products, differ- ences, and quotients) for granted.
SOLUTION 3-6
Let the two rational numbers in question be called rand s. Suppose that the following are true:
r= a/b s= c/d
where a and c are integers, and b and d are nonzero natural numbers. We know such numbers a, b, c, and d exist, because rand sare both rational, and the definition of rational number requires that there exist such numbers a, b, c, and d.
Now consider the mathematical average of the two numbers rand s. We know this number lies between rand s, because the average (or arithmetic mean) of any two numbers is always between them (midway between, in fact!). Call this number x. If we can prove that xis rational, then we have proved the theorem.
We know the following about x, from the formula for finding averages:
x= (r+ s) / 2
That means that this general equation holds true for x:
x= (a/b+ c/d) / 2
From arithmetic, remember the general formula for sums of quotients:
a/b + c/d= (ad+ bc) /bd
Therefore, we know that the following is true: (a/b+ c/d) / 2 = (ad+ bc) /(2bd) and thus:
x= (ad+ bc) / (2bd)
If we can show that the quantity (ad + bc) is an integer and that the quantity 2bdis a nonzero natural number, then we have shown that xis rational. The product of any two integers is always an integer; there- fore adand bcare integers. The sum of any two integers is always an integer; therefore the quantity (ad+ bc) is an integer. The product of any two nonzero natural numbers is a nonzero natural number; there- fore bdis a nonzero natural number. Twice any nonzero natural number is a nonzero natural number; therefore the quantity 2bd is a nonzero natural number. All of this demonstrates that x is equal to an integer divided by a nonzero natural number, and therefore that x is rational. As previously stated, x is the arithmetic mean of r and s, so x lies between rand s. Therefore, for any two distinct rational numbers, there is a third rational number whose value lies between them.
PROBLEM 3-7
Use mathematical induction to show that all natural-number multiples of 0.1 are rational numbers.
SOLUTION 3-7
Remember that the set Nof natural numbers is:
N= {0, 1, 2, 3, 4, . . .}
Therefore, the set Mof natural-number multiples of 0.1 is:
M= {(0 ×0.1), (1 ×0.1), (2 ×0.1), (3 ×0.1), (4 ×0.1), . . .} = {0, 0.1, 0.2, 0.3, 0.4, . . .}
The first element of this set, 0, is rational, because it can be expressed in the form a/b, where ais an integer and bis a nonzero natural number. Simply let a= 0 and b= 1. This is the easy part of the induction proof.
Now for the hard part. Suppose that n×0.1 (which can also be writ- ten as 0.1n) is rational for some unspecified natural number n. Consider the next number in our set Mof multiples, (n+ 1) ×0.1. This can be rearranged using the rules of arithmetic:
(n+ 1) ×0.1 = (n×0.1) + (1 ×0.1) = 0.1n+ 0.1
We know that there exists some integer aand some nonzero natural number b such that 0.1n = a/b, because we are given that 0.1n is rational. Therefore, we can rewrite the above expression as:
0.1n+ 0.1 = a/b+ 0.1 = a/b+ 1/10
Using the arithmetic rule for the sum of two quotients, we can rearrange the above as follows:
a/b+ 1/10 = (10a+ b) /10b
Ten times any integer is another integer; this is a known rule of arithmetic. Therefore, 10a is an integer. The sum of any integer and a nonzero natural number is an integer; this is another rule of arithmetic. Therefore, 10a + b is an integer. Ten times any nonzero natural num- ber is another nonzero natural number; this is yet another rule of arith- metic. Therefore, the quantity (10a + b) /10b is equal to an integer divided by a nonzero natural number. This means, by definition, that (10a + b) /10b is rational. It also happens to be the same quantity as
a/b + 1/10, which in turn is equal to 0.1n+ 0.1, the element immedi- ately after 0.1nin the set M.
We have just proved that if any unspecified element of Mis rational, then the next element is rational as well. That, in addition to the proof that the first element in Mis rational, is all we need to claim that every element in the set Mis rational, based on the principle of mathematical induction.
Quiz
This is an “open book” quiz. You may refer to the text in this chapter. A good score is eight correct. Answers are in the back of the book.
1. Imagine that you want to create an entirely new mathematical theory. The number of postulates in your theory
(a) can be unlimited, and the more the better.
(b) should be large enough so that a contradiction will be easy to derive. (c) should be as small as possible, while still producing a meaningful
theory.
(d) should be zero. You should never assume anything without proof. 2. Consider the following series of statements:
(∀x)(Kx⇒Rx) Kg
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
Rg
This is a generic symbolization of a proof by means of (a) reductio ad absurdum.
(b) mathematical induction. (c) elementary terminology.
(d) straightforward logical deduction.
3. Which of the following symbols is used to denote the fact that an object is an element of a particular set?
(a) ∪ (b) ∩ (c) ∈ (d) ⊄
4. Something that can be described so we have a good idea of what it means, but that is not rigorously defined, is called
(a) an elementary term. (b) an axiom.
(c) a postulate. (d) a lemma.
5. An axiom or postulate is
(a) a fact proved on the basis of other known facts. (b) something assumed to be true without proof.
(c) a minor theorem that follows easily from the proof of a major theorem. (d) a major theorem that is used to prove a minor theorem.
6. Consider the following series of statements:
¬D ⇒(H & ¬Η)
⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯
D
This is a generic symbolization of a proof by means of (a) reductio ad absurdum.
(b) mathematical induction. (c) elementary terminology.
(d) straightforward logical deduction.
7. A mathematical theory can be rendered completely invalid if it is pos- sible to prove, based on its axioms and definitions,
(a) an infinite number of propositions.
(b) only propositions containing existential quantifiers. (c) only propositions containing universal quantifiers. (d) a proposition and also its negation.
8. The set of all objects to which a theorem is intended to apply is called (a) the propositional set.
(b) the predicate set. (c) the empty set. (d) the universe.
9. Suppose you are told that a certain proposition P holds true for some, but not all, rational numbers. You want to prove that P is true for 589/777. The most straightforward, and probably the easiest, way to do this is to demonstrate that P holds true for
(a) 589/777.
(b) all the positive rational numbers.
(c) all the rational numbers between 0 and 1. (d) all the real numbers.
10. Fill in the blank in the following sentence to make it true: “If a proposi- tion P holds true in general for a variable x in a set S, then P is true for any ________ in the set S.”
(a) constant (b) axiom (c) definition (d) corollary