Identifying Errors in Proofs

Identifying errors in purported proofs

Identifying errors in purported proofs. Be aware of common mathematical errors in purported proofs; for example, claiming 'if $ab = ac$, then $b = c$' or assuming 'if $\sin A = \sin B$, then $A = B$' neither of which are valid deductions.

There are lots of pitfalls in setting out proofs and you should start to collect a set of examples of where proofs can go wrong and look out for these sorts of errors and misunderstandings in your own work and in proofs that you are given to study. In this section we shall look at a few examples of the sorts of mistakes and errors that can occur in proofs; but, be warned, this is not an exhaustive list and there are many errors that mathematicians can make when setting out proofs.

Square roots and squaring equations

In this specification we shall take it that $\sqrt{x}$ means the positive number $y$ such that $y^2 = x$; this is standard in mathematics. Generally, we need to be careful with equations when we square them. We need to be careful in case we generate extra solutions to the equation. Here are two examples:

Example 1. Given $x = \sqrt{25}$ [recall this means $x = +5$]:

1. Square both sides: $x^2 = 25$
2. Find all values of $x$ which make $x^2 = 25$ true: $x = \pm 5$

So, we have generated an extra solution, namely $x = -5$ which we didn't have to start with.

Example 2. Find $x$ given $x + 1 = 4$:

1. Square both sides: $(x + 1)^2 = 16$
2. Giving: $x^2 + 2x + 1 = 16$
3. or, $x^2 + 2x - 15 = 0$
4. Factorising: $(x + 5)(x - 3)$
5. Giving solutions: $x = -5$ or $x = +3$

Here we can see that by squaring the original equation we have generated an extra solution, namely $x = -5$.

Exercise Q

$\text{Solve: } \sqrt{2x+3} + \sqrt{x+1} = \sqrt{7x+4}$

Dealing with inequality signs

When students first meet inequality signs, they naturally assume they behave the same way as equals signs: they don't. For an equals sign, the general rule that students go by is that "whatever you do to one side of the equals sign you must do to the other" and this is usually fine for equals signs. However, if you use this rule within inequality signs then it might be the case that the inequality is no longer preserved. Here are some pitfalls that you need to watch out for:

Squaring both sides: $-5 < 4$ is correct but on squaring we obtain the false $25 < 16$.

Multiplying both sides by a negative number: $1 < 2$ is true, but on multiplying by $-1$ we obtain the false $-1 < -2$.

Taking some function of both sides: $\frac{\pi}{4} < \frac{\pi}{3}$ is true but on taking the cosine of both sides we obtain the false $\cos \frac{\pi}{4} < \cos \frac{\pi}{3}$.

Exercise R

1. Given $x < y$ what positive integer values of $n$ make $x^n < y^n$ true?
2. What general characteristics would the function $f$ need to have to make $f(x) < f(y)$ given $x < y$? [Discuss this with others in your class.]
3. Starting with $\frac{x+2}{2x+7} < 5$, is it then valid to deduce $x + 2 < 5(2x + 7)$? Justify your answer. [Discuss this with others in your class.]