Chapter 2 A little logic
One of the main things we are trying to do in mathematics is prove that a statement is always true. A simple example of this is the sentence
The square of an even number is even.
More generally we might try to show that
- a particular mathematical object has some interesting properties,
- when an object has property 1 then it always has property 2,
- an object always has property 3 or property 4 but not both at the same time, or
- you cannot find an object that has property 5.
Let’s spend a little time on this simple example of squaring even numbers and explore why it is true.
- A number \(n\) is even when we can write it as \(n=2k\) where \(k\) is an integer.
- This tells us that the square of that number is \(n^2 = (2k)^2 = 4k^2\text{.}\)
- But now we see that \(n^2\) can be written as two times another number: \(4k^2 = 2(2k^2)\text{.}\)
- And since \(2k^2\) is an integer, we know that \(n^2\) is also even.
Notice that there is quite a lot going on here — a mixture of definitions, language and logic.
Most obviously (we hope) is that we have to understand what even means, so we need to define it. Despite us all being quite familiar with even and odd numbers, we should define it. We should not expect that everyone has exactly the same understanding of even since your readers can come from extremely diverse backgrounds. This author has encountered students who were taught at school that the number \(0\) is neither even nor odd, and others that were taught that only positive numbers can be even or odd. To avoid potential confusion we’ll use the following
A integer is even when it is equal to two times another integer, and an integer is odd when it is not even.
We’ll come back to this definition in the next chapter after we have done a little more logic. We’ll also make it a proper formal definition with bold-text and reference numbers and so on.
Our explanation then consists of sentences asserting bits of mathematics. The sentences are arranged in a particular order and cannot be shuffled around; each one implies the next. To be a good explanation we should take care of our reader and use clear language and enough detail so they can follow along. To make the language flow a little more easily we connect the sentences with words and phrases like “hence”, “this tells us that”, “because of this we can write”. These words and phrases are not just there to make the reader feel a little more comfortable, they also help us to emphasise the logical connections between the sentences to the reader. At the same time, we can expect our reader to do some work; they should be able to understand standard notation, do basic arithmetic and algebra, etc.
In this chapter we won’t do much proving of things, but instead we will focus on basic mathematical sentences and how we combine them together using logic.
