Section 2.1 Statements and open sentences
When you pick up a piece of mathematics you find that it is made up of declarative sentences — sentences that declare something.
The number \(\sqrt{2}\) is not a rational number.
The number 17 is even.
The first sentence is true (we will prove it later in the text) and the second is false. These are sentences that can be assigned a definite truth value — they are either true or false. We will usually denote these \(T\) and \(F\) (and so save ourselves the burden of writing the other 7 characters). A declarative sentence that can be assigned a truth value is called a statement.
The reader will have noticed that the definitions in the previous paragraph are not very formal or precise. Since the authors have been emphasising the importance of being careful and precise, this seems a touch hypocritical 13 . However, giving a precise formal definition of mathematical statement turns out to be quite a lot like the problem of giving a precise formal definition of sets — very difficult and lies beyond the scope of this book 14 . So please excuse the (little) hypocrisy and we’ll just stick with this less formal and more intuitive definition of statement.
Sentences like
I am tall
and
This sentence is false
are not statements since we cannot decide their truth value — they are neither true or false. In the first case it is because we don’t actually know who “I” is. Is it the reader or is it the author? Further, we don’t have a very precise definition of “tall” Indeed the notion of what height constitutes “tall” or “short” can vary dramatically between populations 15 . The second sentence is a little more difficult. If it is true, then it tells us it must be false — how can it be both? While if it is false, then that implies it must be true — again, how can it be both? This sort of self-referential sentence is very difficult to work with, so we will avoid them.
Here are some simple examples of mathematical statements:
- The 100th decimal digit of \(\pi\) is 7.
- The square of the length of the hypotenuse of a right angle triangle is equal to the sum of the squares of the lengths of the other two sides.
- Every even integer greater than \(2\) can be written as the sum of two primes.
The first might be true or might be false, but it must either be true or false 16 . The second is perhaps the most famous theorem any of us know. The last statement is Goldbach’s conjecture and it is not known whether it is true or false. However it is still a statement because it must either be true or it is false.
On the other hand, a sentence like
\begin{equation*}
x^2 - 5x+4 = 0
\end{equation*}
is not a statement because its truth value depends on which number \(x\) we are discussing. In order to assign a truth value we need to know more about \(x\text{.}\) Such sentences are called open sentences. If we assign a value to \(x\) then the open sentence will be either true or false and so become a statement. Usually this variable will come from some predefined set — its “domain”. Often this is the integers or the reals, but typically we should make sure it is clear to the reader. This sentence, \(x^2-5x+4=0\text{,}\) taken over the domain of the integers is true when \(x=1,4\) and otherwise false. We’ll come back to open sentences in Chapter 6.
We now need to start playing with these statements in a more abstract way. This will allow us to talk more generally about doing operations on statements — either operations that act on a single statement or operations that act on pairs of statements. I won’t care too much about the details of the statement (“It is Tuesday” or “I can write with my left-hand”), but rather just its truth value (true and false). So much as we write an integer as \(n\) or \(m\text{,}\) a real number as \(x\) or \(y\text{,}\) I will write a statement as \(P, Q\) or \(R\text{.}\) As for the open sentences, we will use the notation \(P(x),
Q(x), R(x)\text{,}\) since their truth values depend 17 on the value of \(x\text{.}\) This is reminiscent of a function; we put in some value for x and the sentence returns to us a statement. For example, if \(P(x): x^2-5x+4=0\text{,}\) we see that \(P(1)\) is true, while \(P(2)\) is false.
Hypocrisy starts at home.
The interested reader should search-engine their way to articles on the foundations of mathematics, mathematical logic, Richard’s paradox, the Berry paradox, and many other interesting topics that, unfortunately, lie outside the scope of this text (since it must have a finite length).
At the time of typing this author looked up wikipedia to find that the average height of a man in the Netherlands is about 184cm, while in Vietnam it is about 162cm. Quite a sizeable difference. Sorry for the pun.
Actually it is true, though you have to be careful how you count — since the number starts \(3.141\dots\) we have counted the initial “3” as the first decimal digit. On the other hand, if you start counting from the first “1”, then the 100th digit is “9”. Definitions matter.
When the open sentence depends on more than one variable, say \(x, k\) we will write \(P(x,k), Q(x,k)\) and so on.
