Theorem Proving in Mathematics
The very foundation of mathematics as we know it! Or, well, that evil thing math teachers had invented so they could torture the students. And we all know that very well… Anyway, I’m one of the tortured this fall, so let’s talk the math!
Mathematical proving is a way of convincingly demonstrating that some mathematical statement is necessarily true[1]. There are several different types and ways of proofs and proving things, but one thing is common among all of them - a proof can only be obtained from unquestionable truths (axioms) by some sort of deductive reasoning. Experience, intuition or belief are not valid arguments when it comes to mathematical proving.
When you show that your statement works for every possible element from the universe, your statement or proposition becomes a theorem. Now, let’s have a look at some of the techniques that are frequently used to prove stuff.
Direct Proof
Direct proof in mathematics is a way of proving that some statement is true by showing that the statement can be deduced from generally anything that is known to be true (which are some basic truths - axioms and all previously proven theorems). It’s most common for proving conditional statements in a form of A → B.
Formally speaking, when constructing a direct proof, were looking for a sequence of statements A1, A2, …, An so (A1 → A2) ∧ … ∧ (An-1 → An) ∧ T, where T = A → B is the statement we’re proving.
It sounds weird, but it’s really simple. Take for instance solving an equation. Almost any 5th grader can do that, right? If you think about that, the sequence of steps you go through in order to get to the value of x is a sort of a direct proof. You have some precondition (the equation) and some expected result (e.g., x = 5). Also there are some basic axioms that your math teacher told you (like subtracting 5 of both sides of the equation). While solving the equation, you try to come up with the right sequence of steps that will get you from x + 1 - 2 x = -4 to x = 5.
The solution looks like this:
- x + 1 - 2 x = -4 →
- x - 2 x = -5 →
- -x = -5 →
- x = 5
We have proven that with the given precondition, x = 5. Formally speaking, x + 1 - 2 x = -4 → x = 5.
Proof by Induction
More precisely proof by mathematical induction. This proof is often used when we need to show, that something is true for an infinitely large universum (natural numbers for instance). It’s tied to the universal quantification in predicate logic.
The proof consists of two steps. At first, base case is proved for some element from the universe, then an induction rule is used to prove all other cases.
- The basis (base case): showing that the statement holds when n is equal to the lowest value that n is given in the question. Usually, n = 0 or n = 1.
- The inductive step: showing that if the statement holds for some n, then the statement also holds when n + 1 is substituted for n.
The principle can be illustrated on the ” domino effect” on a row of falling dominoes.
- Base case - The first domino will fall.
- Induction rule - Whenever a domino falls, its next neighbor will also fall.
Proof by Contradiction
Proof by contradiction is rather common as well. In this proof, it is shown that if some statement were so, a logical contradiction occurs, hence the statement must be not so. It starts by adding a statement to the preconditions, that we expect to be false. Then we try to show why the precondition is so, but in the process we find some contradiction.
Textbook case of the proof by contradiction is proving that a language is not regular by pumping theorem. We say, that a language is regular then pumping lemma states, that if a language is regular, some conditions and restrictions must be met.
Here’s an example proof from wikipedia[1]:
Suppose that sqrt(2) wes a rational number, so by definition sqrt(2) = a/b where a and b are non-zero integers with no common factor. Thus, b sqrt(2) = a. Squaring both sides yields 2 b 2 = a 2. Since 2 divides the left hand side, 2 must also divide the right hand side (as they are equal and both integers). So a 2 is even, which implies that a must also be even. So we can write a = 2 c, where c is also an integer. Substitution into the original equation yields 2_b_2 = (2_c_)2 = 4 _c_2. Dividing both sides by 2 yields b 2 = 2 c 2. But then, by the same argument as before, 2 divides b 2, so b must be even. However, if a and b are both even, they share a factor, namely 2. This contradicts our assumption, so we are forced to conclude that sqrt(2) is an irrational number.
At first we make an assumption which is followed by sequence of steps, that should be valid, if the former statement is so until we find a contradiction, therefore the opposite is true.
Summary
Proofs in mathematics can be a little… eh, well, fricking intimidating at first. They’re strictly formal, sometimes very hard to understand and it might seem almost impossible to come up with one, when you need to. Proving theorems requires a lot of knowledge and experience. Proofs in math might not be for anybody, but it’s good to know they’re there :-).