12.2: Proofs

Logic plays a basic role in human affairs. Scientists use logic to draw conclusions from experiments, judges use it to deduce consequences of the law, and mathematicians use it to prove theorems. Logic arises in ordinary speech with assertions such as “If John studies hard, he will pass the course,” or “If an integer \(n\) is divisible by \(6\), then \(n\) is divisible by \(3\).”¹ In each case, the aim is to assert that if a certain statement is true, then another statement must also be true. In fact, if \(p\) and \(q\) denote statements, most theorems take the form of an implication: “If \(p\) is true, then \(q\) is true.” We write this in symbols as

\[p \Rightarrow q \nonumber \]

and read it as “\(p\) implies \(q\).” Here \(p\) is the hypothesis and \(q\) the conclusion of the implication. The verification that \(p \Rightarrow q\) is valid is called the proof of the implication. In this section we examine the most common methods of proof² and illustrate each technique with some examples.

Method of Direct Proof

To prove that \(p \Rightarrow q\), demonstrate directly that \(q\) is true whenever \(p\) is true.

034634 If \(n\) is an odd integer, show that \(n^{2}\) is odd.

If \(n\) is odd, it has the form \(n = 2k + 1\) for some integer \(k\). Then \(n^{2} = 4k^{2} + 4k + 1 = 2(2k^{2} + 2k) + 1\) also is odd because \(2k^{2} + 2k\) is an integer.

Note that the computation \(n^{2} = 4k^{2} + 4k + 1\) in Example [exa:034634] involves some simple properties of arithmetic that we did not prove. These properties, in turn, can be proved from certain more basic properties of numbers (called axioms)—more about that later. Actually, a whole body of mathematical information lies behind nearly every proof of any complexity, although this fact usually is not stated explicitly. Here is a geometrical example.

034648 In a right triangle, show that the sum of the two acute angles is 90 degrees.

The right triangle is shown in the diagram. Construct a rectangle with sides of the same length as the short sides of the original triangle, and draw a diagonal as shown. The original triangle appears on the bottom of the rectangle, and the top triangle is identical to the original (but rotated). Now it is clear that \(\alpha + \beta\) is a right angle.

Geometry was one of the first subjects in which formal proofs were used—Euclid’s Elements was published about 300 B.C. The Elements is the most successful textbook ever written, and contains many of the basic geometrical theorems that are taught in school today. In particular, Euclid included a proof of an earlier theorem (about 500 B.C.) due to Pythagoras. Recall that, in a right triangle, the side opposite the right angle is called the hypotenuse of the triangle.

Pythagoras’ Theorem034656

In a right-angled triangle, show that the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides.

Let the sides of the right triangle have lengths \(a\), \(b\), and \(c\) as shown. Consider two squares with sides of length \(a + b\), and place four copies of the triangle in these squares as in the diagram. The central rectangle in the second square shown is itself a square because the angles \(\alpha\) and \(\beta\) add to \(90\) degrees (using Example [exa:034648]), so its area is \(c^{2}\) as shown. Comparing areas shows that both \(a^{2} + b^{2}\) and \(c^{2}\) each equal the area of the large square minus four times the area of the original triangle, and hence are equal.

Sometimes it is convenient (or even necessary) to break a proof into parts, and deal with each case separately. We formulate the general method as follows:

Method of Reduction to Cases

To prove that \(p \Rightarrow q\), show that \(p\) implies at least one of a list \(p_{1}, p_{2}, \dots , p_{n}\) of statements (the cases) and then show that \(p_{i} \Rightarrow q\) for each \(i\).

034677 Show that \(n^{2} \geq 0\) for every integer \(n\).

This statement can be expressed as an implication: If \(n\) is an integer, then \(n^{2} \geq 0\). To prove it, consider the following three cases:

\[(1) \ n > 0; \quad (2) \ n = 0; \quad (3) \ n < 0. \nonumber \]

Then \(n^{2} > 0\) in Cases (1) and (3) because the product of two positive (or two negative) integers is positive. In Case (2) \(n^{2} = 0^{2} = 0\), so \(n^{2} \geq 0\) in every case.

034691 If \(n\) is an integer, show that \(n^{2} - n\) is even.

We consider two cases:

\[(1) \ n \mbox{ is even; } \quad (2) \ n \mbox{ is odd.} \nonumber \]

We have \(n^{2} - n = n(n - 1)\), so this is even in Case (1) because any multiple of an even number is again even. Similarly, \(n - 1\) is even in Case (2) so \(n(n - 1)\) is again even for the same reason. Hence \(n^{2} - n\) is even in any case.

The statements used in mathematics are required to be either true or false. This leads to a proof technique which causes consternation in many beginning students. The method is a formal version of a debating strategy whereby the debater assumes the truth of an opponent’s position and shows that it leads to an absurd conclusion.

Method of Proof by Contradiction

To prove that \(p \Rightarrow q\), show that the assumption that both \(p\) is true and \(q\) is false leads to a contradiction. In other words, if \(p\) is true, then \(q\) must be true; that is, \(p \Rightarrow q\).

034707 If \(r\) is a rational number (fraction), show that \(r^{2} \neq 2\).

To argue by contradiction, we assume that \(r\) is a rational number and that \(r^{2} = 2\), and show that this assumption leads to a contradiction. Let \(m\) and \(n\) be integers such that \(r = \frac{m}{n}\) is in lowest terms (so, in particular, \(m\) and \(n\) are not both even). Then \(r^{2} = 2\) gives \(m^{2} = 2n^{2}\), so \(m^{2}\) is even. This means \(m\) is even (Example [exa:034634]), say \(m = 2k\). But then \(2n^{2} = m^{2} = 4k^{2}\), so \(n^{2} = 2k^{2}\) is even, and hence \(n\) is even. This shows that \(n\) and \(m\) are both even, contrary to the choice of these numbers.

Pigeonhole Principle034724 If \(n + 1\) pigeons are placed in \(n\) holes, then some hole contains at least \(2\) pigeons.

Assume the conclusion is false. Then each hole contains at most one pigeon and so, since there are \(n\) holes, there must be at most \(n\) pigeons, contrary to assumption.

The next example involves the notion of a prime number, that is an integer that is greater than 1 which cannot be factored as the product of two smaller positive integers both greater than \(1\). The first few primes are \(2, 3, 5, 7, 11, \dots\).

034732 If \(2^n - 1\) is a prime number, show that \(n\) is a prime number.

We must show that \(p \Rightarrow q\) where \(p\) is the statement “\(2^n - 1\) is a prime”, and \(q\) is the statement “\(n\) is a prime.” Suppose that \(p\) is true but \(q\) is false so that \(n\) is not a prime, say \(n = ab\) where \(a \geq 2\) and \(b \geq 2\) are integers. If we write \(2^a = x\), then \(2^n = 2^{ab} = (2^a)^b = x^b\). Hence \(2^n - 1\) factors:

\[2^n - 1 = x^b - 1 = (x-1)(x^{b-1} +x^{b-2} + \cdots + x^2 + x + 1 ) \nonumber \]

As \(x \geq 4\), this expression is a factorization of \(2^n - 1\) into smaller positive integers, contradicting the assumption that \(2^n - 1\) is prime.

The next example exhibits one way to show that an implication is not valid.

034752 Show that the implication “\(n\) is a prime \(\Rightarrow 2^n - 1\) is a prime” is false.

The first four primes are \(2\), \(3\), \(5\), and \(7\), and the corresponding values for \(2^n - 1\) are \(3\), \(7\), \(31\), \(127\) (when \(n = 2\), \(3\), \(5\), \(7\)). These are all prime as the reader can verify. This result seems to be evidence that the implication is true. However, the next prime is \(11\) and \(2^{11} - 1 = 2047 = 23 \cdot 89\), which is clearly not a prime.

We say that \(n = 11\) is a counterexample to the (proposed) implication in Example [exa:034752]. Note that, if you can find even one example for which an implication is not valid, the implication is false. Thus disproving implications is in a sense easier than proving them.

The implications in Example [exa:034732] and Example [exa:034752] are closely related: They have the form \(p \Rightarrow q\) and \(q \Rightarrow p\), where \(p\) and \(q\) are statements. Each is called the converse of the other and, as these examples show, an implication can be valid even though its converse is not valid. If both \(p \Rightarrow q\) and \(q \Rightarrow p\) are valid, the statements \(p\) and \(q\) are called logically equivalent. This is written in symbols as

\[p \Leftrightarrow q \nonumber \]

and is read “\(p\) if and only if \(q\)”. Many of the most satisfying theorems make the assertion that two statements, ostensibly quite different, are in fact logically equivalent.

034764 If \(n\) is an integer, show that “\(n\) is odd \(\Leftrightarrow n^{2}\) is odd.”

In Example [exa:034634] we proved the implication “\(n\) is odd \(\Rightarrow n^{2}\) is odd.” Here we prove the converse by contradiction. If \(n^{2}\) is odd, we assume that \(n\) is not odd. Then \(n\) is even, say \(n = 2k\), so \(n^{2} = 4k^{2}\), which is also even, a contradiction.

Many more examples of proofs can be found in this book and, although they are often more complex, most are based on one of these methods. In fact, linear algebra is one of the best topics on which the reader can sharpen his or her skill at constructing proofs. Part of the reason for this is that much of linear algebra is developed using the axiomatic method. That is, in the course of studying various examples it is observed that they all have certain properties in common. Then a general, abstract system is studied in which these basic properties are assumed to hold (and are called axioms). In this system, statements (called theorems) are deduced from the axioms using the methods presented in this appendix. These theorems will then be true in all the concrete examples, because the axioms hold in each case. But this procedure is more than just an efficient method for finding theorems in the examples. By reducing the proof to its essentials, we gain a better understanding of why the theorem is true and how it relates to analogous theorems in other abstract systems.

The axiomatic method is not new. Euclid first used it in about 300 B.C. to derive all the propositions of (euclidean) geometry from a list of 10 axioms. The method lends itself well to linear algebra. The axioms are simple and easy to understand, and there are only a few of them. For example, the theory of vector spaces contains a large number of theorems derived from only ten simple axioms.

Exercises for [chap:appbproofs]

solutions

2

In each case prove the result and either prove the converse or give a counterexample.

1. If \(n\) is an even integer, then \(n^{2}\) is a multiple of \(4\).
2. If \(m\) is an even integer and \(n\) is an odd integer, then \(m + n\) is odd.
3. If \(x = 2\) or \(x = 3\), then \(x^{3} - 6x^{2} + 11x - 6 = 0\).
4. If \(x^{2} - 5x + 6 = 0\), then \(x = 2\) or \(x = 3\).

2. If \(m = 2p\) and \(n = 2q + 1\) where \(p\) and \(q\) are integers, then \(m + n = 2(p + q) + 1\) is odd. The converse is false: \(m = 1\) and \(n = 2\) is a counterexample.
3. \(x^{2} - 5x + 6 = (x - 2)(x - 3)\) so, if this is zero, then \(x = 2\) or \(x = 3\). The converse is true: each of \(2\) and \(3\) satisfies \(x^{2} - 5x + 6 = 0\).

In each case either prove the result by splitting into cases, or give a counterexample.

1. If \(n\) is any integer, then \(n^{2} = 4k + 1\) for some integer \(k\).
2. If \(n\) is any odd integer, then \(n^{2} = 8k + 1\) for some integer \(k\).
3. If \(n\) is any integer, \(n^{3} - n = 3k\) for some integer \(k\). [Hint: Use the fact that each integer has one of the forms \(3k\), \(3k + 1\), or \(3k + 2\), where \(k\) is an integer.]

2. This implication is true. If \(n = 2t + 1\) where \(t\) is an integer, then \(n^{2} = 4t^{2} + 4t + 1 = 4t(t + 1) + 1\). Now \(t\) is either even or odd, say \(t = 2m\) or \(t = 2m + 1\). If \(t = 2m\), then \(n^{2} = 8m(2m + 1) + 1\); if \(t = 2m + 1\), then \(n^{2} = 8(2m + 1)(m + 1) + 1\). Either way, \(n^{2}\) has the form \(n^{2} = 8k + 1\) for some integer \(k\).

In each case prove the result by contradiction and either prove the converse or give a counterexample.

1. If \(n > 2\) is a prime integer, then \(n\) is odd.
2. If \(n + m = 25\) where \(n\) and \(m\) are integers, then one of \(n\) and \(m\) is greater than \(12\).
3. If \(a\) and \(b\) are positive numbers and \(a \leq b\), then \(\sqrt{a} \leq \sqrt{b}\).
4. If \(m\) and \(n\) are integers and \(mn\) is even, then \(m\) is even or \(n\) is even.

2. Assume that the statement “one of \(m\) and \(n\) is greater than \(12\)” is false. Then both \(n \leq 12\) and \(m \leq 12\), so \(n + m \leq 24\), contradicting the hypothesis that \(n + m = 25\). This proves the implication. The converse is false: \(n = 13\) and \(m = 13\) is a counterexample.
3. Assume that the statement “\(m\) is even or \(n\) is even” is false. Then both \(m\) and \(n\) are odd, so \(mn\) is odd, contradicting the hypothesis. The converse is true: If \(m\) or \(n\) is even, then \(mn\) is even.

Prove each implication by contradiction.

1. If \(x\) and \(y\) are positive numbers, then \(\sqrt{x + y} \neq \sqrt{x} + \sqrt{y}\).
2. If \(x\) is irrational and \(y\) is rational, then \(x + y\) is irrational.
3. If \(13\) people are selected, at least \(2\) have birthdays in the same month.

2. If \(x\) is irrational and \(y\) is rational, assume that \(x + y\) is rational. Then \(x = (x + y) - y\) is the difference of two rationals, and so is rational, contrary to the hypothesis.

Disprove each statement by giving a counterexample.

1. \(n^{2} + n + 11\) is a prime for all positive integers \(n\).
2. \(n^{3} \geq 2^n\) for all integers \(n \geq 2\).
3. If \(n \geq 2\) points are arranged on a circle in such a way that no three of the lines joining them have a common point, then these lines divide the circle into \(2^{n-1}\) regions. [The cases \(n = 2\), \(3\), and \(4\) are shown in the diagram.]

2. \(n = 10\) is a counterexample because \(10^3 = 1000\) while \(2^{10} = 1024\), so the statement \(n^{3} \geq 2^n\) is false if \(n = 10\). Note that \(n^{3} \geq 2^n\) does hold for \(2 \leq n \leq 9\).

The number \(e\) from calculus has a series expansion

\[e = 1 + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \cdots \nonumber \]

where \(n! = n(n - 1) \cdots 3 \cdot 2 \cdot 1\) for each integer \(n \geq 1\). Prove that \(e\) is irrational by contradiction. [Hint: If \(e = m/n\), consider

\[k = n! \left(e- 1 - \frac{1}{1!} - \frac{1}{2!} - \frac{1}{3!} - \cdots - \frac{1}{n!} \right). \nonumber \]

Show that \(k\) is a positive integer and that

\[k = \frac{1}{n+1} + \frac{1}{(n+1)(n+2)} + \cdots < \frac{1}{n}. ] \nonumber \]

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1. By an integer we mean a “whole number”; that is, a number in the set \(0, \pm 1, \pm 2, \pm 3, \dots\)↩
2. For a more detailed look at proof techniques see D. Solow, How to Read and Do Proofs, 2nd ed. (New York: Wiley, 1990); or J. F. Lucas. Introduction to Abstract Mathematics, Chapter 2 (Belmont, CA: Wadsworth, 1986).↩