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What Is a Mathematical Proof?

In math, being able to prove what you are doing is of great importance. A proof is a string of implications and equivalences, where the entire text is the answer. In a regular mathematical problem, you often draw two lines beneath your last expression to show that you have reached a final answer. That is unnecessary in a proof since the answer is the whole text.

Instead, proofs often end with the abbreviation Q.E.D. This is Latin and stands for “quod erat demonstrandum”, which means “which was to be demonstrated”. Another symbol that is used to show that a proof is complete is the “Halmos symbol”. It looks like this: $□$ or $■$

There are many types of proofs. The ones you can lean more about on this website are: Direct proofs, proofs by contrapositive, proofs by induction, and one proof of the Pythagorean theorem. But first, a little bit about proofs in general.

A proof is an explanation for why something is true. What you are trying to prove is a conclusion. Here is a simple proof:

Assumption 1: Every human is mortal.
Assumption 2: Socrates is a human.
Conclusion: Socrates is mortal.

Every proof builds on one or more assumptions. The proof has two assumptions: “Every human is mortal” and “Socrates is a human”. The proof doesn’t need to convince you that the assumptions are true—you are allowed to think that Socrates is an undercover zebra or that humans can become immortal if they don’t eat carbohydrates. But, if you accept the assumptions, the proof explains why you also have to accept that the conclusion is true.

Theory

Proofs in Mathematics

When you are working with a proof, you always want to start with an assumption, and end with a desired conclusion, by using logical steps. One step should always follow from the previous—every step should be an implication or an equivalence.

Example 1

Prove that if $n$ is an even number, then $n^{2}$ is an even number:

The assumption here is that you have a number $n$ that is an even number—that is, it is 2 times a number, or said differently, that $n = 2 m$ for an integer $m$ . Then, you see that

\begin{array}{l} n & = 2 m \\ ⇓ \\ n^{2} & = {(2 m)}^{2} \\ = 4 m^{2} \\ = 2 (2 m^{2}) \\ = 2 \cdot k, where k = 2 m^{2} \end{array}

\begin{array}{l} n = 2 m \Rightarrow n^{2} & = {(2 m)}^{2} \\ = 4 m^{2} \\ = 2 (2 m^{2}) \\ = 2 \cdot k, where k = 2 m^{2} \end{array}

n^{2}

is also 2 times a number, and therefore an even number. This was what you wanted to conclude with.

Q.E.D

Example 2

Prove that among three subsequent integers, one of them is divisible by 3:

You know that all the numbers in the third row of the times tables ( $3, 6, 9, 12, \dots$ ) are divisible by 3. From this you can conclude that every third subsequent integer is divisible by 3, and that there are exactly two integers between them that are not divisible by 3.

That means that among the three subsequent integers, one of them must be a multiple of 3. The logic goes as follows: If none of the three subsequent numbers were multiples of 3, one of the numbers that are multiples of 3 would be missing. That can’t be correct, so one of the three subsequent numbers must be divisible by 3.

Q.E.D

Here is another way to prove the same thing:

Example 3

Prove that among three subsequent integers, one of them is divisible by 3:

You know that subsequent integers can be expressed like this: You call an arbitrary integer $n$ . Then, the number that is one larger is $n + 1$ and the number that is one larger than that again is $n + 2$ . The subsequent numbers are then:

n, n + 1, n + 2

When you try to divide $n$ by 3, there are three possibilities: You can get 0, 1 or 2 as the remainder (remainder = what is left when a division doesn’t fully go into the divisor). Thus, you have the following possibilities:

1.

If you have 0 as the remainder, it means that the division went okay, so

n

is divisible by 3.

2.

If you get 1 as the remainder, it means that

n - 1

is divisible by 3, because

n - 1 + 1 = n

. In this case,

(n - 1) + 3 = n + 2

is also divisible by 3, since every third integer is divisible by 3.

3.

If you get 2 as the remainder, it means that

n - 2

is divisible by 3, because

n - 2 + 2 = n

. In this case,

(n - 2) + 3 = n + 1

is also divisible by 3, since every third integer is divisible by 3.

Therefore, you see that in each case, one of the numbers $n$ , $n + 1$ or $n + 2$ is divisible by 3. Therefore, among three subsequent integers, there is always a number that is divisible by 3.

Q.E.D

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What Is the Difference Between Equivalence and Implication?

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