2.4 Our first proofs

In this section we give some simple proofs to illustrate the ideas discussed so far.

Claim 2.16.

There is an integer strictly between $3.6$ and $5.7$ on the real number line.

This statement uses an existential quantifier phrased as ‘there is’. To prove a statement with an existential quantifier, you either have to find a mathematical object with the desired property (this is known as a constructive proof), or prove that such an object must exist without finding it explicitly (this is known as a non-constructive proof). You do not have to determine how many such objects exists, just that there is at least one.

Proof.

The integer $4$ is strictly between $3.6$ and $5.7$ . ∎

Proofs are typically laid out like this, with a clear indication of where the proof starts and a clear indication of where it ends. The symbol ∎ is called a halmos and it is commonly used to indicate the end of a proof.

Claim 2.16 is not supposed to be difficult to prove, and you might even think that the claim is so obvious that it does not require a proof. You will experience this phenomenon often when studying mathematics – sometimes things that your lecturers or classmates think are obvious might not be obvious to you, and the other way around.

Note that there is an alternative proof where we give $5$ as an integer strictly between $3.6$ and $5.7$ . This is perfectly good too. In general, a statement has many correct proofs.

Claim 2.17.

Given any even integer $a$ , the integer $a+4$ is also even.

This statement uses a universal quantifier phrased as ‘given any’. To prove a statement with a universal quantifier, you must not assume that the objects involved have any additional properties other than those in the statement. For example, assuming that $a=2$ in Claim 2.17 and concluding that $2+4$ is even does not prove the claim.

Proof.

By Definition 2.6, we can write $a=2k$ for some integer $k$ . Then

a+4=2k+4=2(k+2).

As $k$ is an integer, so is $(k+2)$ . Therefore $a+4$ fits the definition of an even integer. ∎

Note that proofs are written in complete sentences of English. A proof should communicate an argument to whoever is the intended reader. It can be useful to read a finished proof aloud to make sure it is written in clear and complete sentences.

Claim 2.18.

Given any two even integers $a$ and $b$ , their sum $a+b$ is also even.

Proof.

Suppose that $a$ and $b$ are even integers. By Definition 2.6, we can write $a=2k$ and $b=2l$ for some integers $k$ and $l$ . Then

a+b=2k+2l=2(k+l).

As $k$ and $l$ are integers then so is $(k+l)$ . Therefore $a+b$ fits the definition of an even integer. ∎

Exercise 2.19.

Prove that the sum of any two odd integers is even.

Proof.

Let $a$ and $b$ be two odd integers. From the definitions in Section 2.2.2, we can therefore write $a=2k+1$ and $b=2l+1$ for some integers $k$ and $l$ .

We then consider the sum

	$\displaystyle a+b$	$\displaystyle=(2k+1)+(2l+1)$
		$\displaystyle=2k+2l+2$
		$\displaystyle=2(k+l+1).$

As $k+l+1$ is an integer, we conclude that the sum $a+b$ must be even. ∎

There are usually many different ways of proving a result. We demonstrate this by discussing an example.

Example 2.20.

Recall Claim 2.8: for all positive real numbers $x$ , $x^{2}+x$ is positive. This claim is, in fact, true and we present some proofs below.

Observe that the claim involves a universal quantifier and so we cannot reason by thinking about specific real numbers. Here is one possible proof.

Proof 1. Since $x$ is a positive real number, we know that $x^{2}$ is also a positive real number. Thus the expression $x^{2}+x$ is the sum of two positive real numbers. Adding together two positive real numbers always gives another positive real number and so $x^{2}+x$ must be positive. ∎

We could use more symbols and present a shorter argument, but we should still write in complete sentences.

Proof 2. Since $x>0$ , it follows that $x^{2}>0$ too. The sum of two positive real numbers is positive, therefore $x^{2}+x>0$ . ∎

Or we could make a different argument.

Proof 3. First observe that $x^{2}+x=x(x+1)$ . Because $x$ is positive then so is $x+1$ . The product of two positive real numbers is also positive, therefore $x(x+1)>0$ . ∎

Or you might think about this problem in a different way.

Proof 4. First observe that $x^{2}+x=x(x+1)$ . The expression $x(x+1)$ is positive precisely when the curve $y=x(x+1)$ lies above the $x$ -axis.

From the graph, we can see that $x^{2}+x>0$ when $x>0$ . ∎

This last proof leaves some important facts unstated, assuming that the reader can fill these gaps using their wider mathematical knowledge. For instance, it is important that the curve is continuous and that it has no other turning points further to the right of the sketch.

You may prefer one of these proofs over the others, but all are correct.

Exercise 2.21.

For each of Claims 2.9, 2.10 and 2.12, decide if the claim is true and, if so, try to write out a proof.

Remember that writing a proof is not the same as doing rough working for your own understanding. You should write an argument, in complete sentences, to be understood by your reader.

Solution.Claim 2.9: ‘For all real numbers $x$ , $x^{2}+x$ is positive.’ This is false; for example, if $x=0$ then $x^{2}+x=0$ , which is not positive.

There is further discussion of disproving statements in Section 4.1.

Claim 2.10: ‘For all real numbers $x$ strictly between $-1$ and $0$ , $x^{2}+x$ is negative.’ This is true.

Proof.

We can rewrite $x^{2}+x$ as $x(x+1)$ . For $x$ strictly between $-1$ and $0$ , $(x+1)$ is positive. Thus $x(x+1)$ is the product of a negative and positive real number, and so must be negative. ∎

Or you may prefer to reason graphically:

Proof.

First observe that $x^{2}+x=x(x+1)$ . The expression $x(x+1)$ is negative precisely when the curve $y=x(x+1)$ lies below the $x$ -axis.

From the graph, we can see that $x^{2}+x<0$ when $-1<x<0$ . ∎

Claim 2.12: ‘There exists a real number $x$ such that $x^{2}+x$ is positive.’ This is true.

Proof.

Take $x=1$ then $x^{2}+x=1^{2}+1=2$ , which is positive. ∎

Any real number less than $-1$ or greater than $0$ also has this property.