This brings us to the heart of this chapter. A mathematical proof starts with facts that you already know to be true and combines them in a series of valid arguments that reaches a new or interesting conclusion. The following argument forms are valid and used with enough frequency that they have their own names. You should confirm for yourself that these arguments are valid and their are additional notes by the more common ones.
Modus Ponens
Modus Tollens
Transitivity
p → q p q
p → q ~q ~p
p → q q → r p → r
Elimination (1)
Elimination (2)
The Contradiction Rule
p ∨ q ~q p
p ∨ q ~p q
~p → c p
Generalization (1)
Generalization (2)
Conjunction
pp ∨ q
qp ∨ q
~p ∨ q
~qp
Proof by Cases
Specialization (1)
Specialization (2)
p ∨ q p → r q → rr
p ∧ qp
p ∧ qq
Logical Fallacies (Invalid Arguments)
It's as important to know what you can't do as it is to know what you can. These areguments are sometimes asserted by students but they don't always result in valid conslusions.
Fallacy of the Converse
Fallacy of the Inverse
p → q q p
p → q ~p ~q
Module Ponens
Literally, "the method of pulling by placing", this is one of the most common arguments used in mathematical proofs. It lets us apply a general rule to a specific object. For example: If a quadrilateral has four right angles then it's a rectangle. Quadrilateral ABCD has four right angles. Therefore, ABCD is a rectangle.
You might have noticed that the antecedant in the theorem didn't exactly match the "ABCD has four right angles" part but remember that the conditional statement is a statement about all quadrilaterals. To make it a more precise match, it could be rephrased, "If ABCD is a quadrilateral with four right angles then it's a rectangle."
Module Tollens
Also known as "denying the conesquent", this literally means "the method of removing by taking away". Notice how this is similar to the contrapositive, ~q → ~p. In practice, this is saying that if we know the consequence is not true then the antecedant must also not be true. For example, if a quadrilateral has four right angles then it's a rectangle. Quadrilateral ABCD is not a rectangle. Therefore, ABCD does not have four right angles.
The Contradiction Rule
This is the basis for a method of proof called "proof by contradiction". The idea here is that if a statement leads to a contradiction then the statement must be false. We'll talk more about proof by contradiction later in the class but the idea is that you start by assuming that your premise is false then show that that assumption leads to a contradiction, i.e. a false conclusion.
The Fallacy of the Converse
Remember from our previous discussions that a conditional statement and its converse aren't equivalent. That's what this argument is (incorrectly) asserting. For example, the argument, "If a function is differentiable then it's continuous. f is continuous, therefore it's differentiable." is invalid, i.e. the conclusion is false even if the premises are true. For example, $f(x) = |x|$ is continuous everywhere but it isn't differentiable at 0.
The Fallacy of the Inverse
Remember from our previous discussions that a conditional statement and its inverse aren't equivalent. That's what this argument is (incorrectly) asserting. For example, the argument, "If a function is differentiable then it's continuous. f is not differentiable, therefore it's not continuous." is invalid, i.e. the conclusion is false even if the premises are true. For example, $f(x) = |x|$ is not differentiable at 0 but it is continuous everywhere.
Elimination
This is popluarly referred to as the "process of elimination". Recall that for an "or" statement to be true, at least one of its components has to be true. If we know that one is false then that forces the other one to be true.
