3: Proof Techniques I
Love is a snowmobile racing across the tundra and then suddenly it flips over, pinning you underneath. At night, the ice weasels come.
–Matt Groening
-
- 3.1: Direct Proofs of Universal Statements
- Generally, the first thing to do in proving a universal statement is to rephrase it as a conditional. The resulting statement is a Universal Conditional Statement or a UCS. The reason for taking this step is that the hypotheses will then be clear – they form the antecedent of the UCS. So, while you won’t have really made any progress in the proof by taking this advice, you will at least know what tools you have at hand.
-
- 3.2: More Direct Proofs
- In creating a direct proof, we need to look at our hypotheses, consider the desired conclusion, and develop a strategy for transforming A into B. Quite often you’ll find it easy to make several deductions from the hypotheses, but none of them seems to be headed in the direction of the desired conclusion. The usual advice at this stage is “Try working backwards from the conclusion.”
-
- 3.3: Indirect Proofs- Contradiction and Contraposition
- Indirect proofs takes a completely different tack from direct proofs. If we are trying to prove that all thrackles are polycyclic, we will begin by supposing that you had a thrackle that wasn’t polycyclic, and furthermore, show that this supposition leads to something truly impossible. Well, if it’s impossible for a thrackle to not be polycyclic, then it must be the case that all of them are. Such an argument is known as proof by contradiction.
-
- 3.4: Disproofs
- The idea of a “disproof” is really just semantics – in order to disprove a statement we need to prove its negation. If we are given a universally quantified statement the first thing to do is try it out for some random elements of the universe we’re working in. If we happen across a value that satisfies the statement’s hypotheses but doesn’t satisfy the conclusion, we’ve found what is known as a counterexample.
-
- 3.5: Even More Direct Proofs- By Cases and By Exhaustion
- Proof by exhaustion is the least attractive proof method from an aesthetic perspective. An exhaustive proof consists of literally (and exhaustively) checking every element of the universe to see if the given statement is true for it. Usually, of course, this is impossible because the universe of discourse is infinite; but when the universe of discourse is finite, one certainly can’t argue the validity of an exhaustive proof.
-
- 3.6: Proofs and Disproofs of Existential Statements
- From a certain point of view, there is no need for the current section. If we are proving an existential statement we are disproving some universal statement. (Which has already been discussed.) Similarly, if we are trying to disprove an existential statement, then we are actually proving a related universal statement. Nevertheless, sometimes the way a theorem is stated emphasizes the existence question over the corresponding universal.