# I Book involving proofs

1. Nov 11, 2016

### Mr Davis 97

My textbook goes into depth about proof techniques and about how to go about proving theorems. However, the author only really focuses on theorems that are stated in the form "if p, then q." I know that a great many theorems have this logical structure, so it is good to know how to prove them, using direct, contrapositive, and contradiction techniques. However, what if a theorem does not have this "if p, then q" structure? What if it is just stated as a fact, p? How are these types of statements proved in general? We can't use a direct proof, because we don't have a hypothesis, and we can't use contrapositive because it is not in the conditional form. Can we only prove it using definitions and/or contradiction?

2. Nov 11, 2016

### Staff: Mentor

"if p, then q" is a fact "r", and every such statement can be seen as "if 1=1, then r". The categories you want to make do not exist.

3. Nov 11, 2016

### Staff: Mentor

Facts in mathematics are axioms. It's only important that a set of axioms is without a contradiction.
What you might have in mind to be a fact very likely depends on some given definitions. E.g. a statement "2 is an even number." seems to be a fact, but it depends on the definition of even numbers and could be stated: "Even numbers are divisible by 2. Then 2 is an even number." If you define even numbers as those, which are not divisible by 2, then 2 is not an even number. Both is possible. The fact that it would be unusual doesn't make it wrong as a logic statement.

4. Nov 11, 2016

### Mr Davis 97

Well there are different methods of proof depending on what form the proposition is in, right?

5. Nov 11, 2016

### Hercuflea

Check out the first 5 chapters of "an interactive introduction to mathematical analysis" by jonathan Lewin

6. Nov 11, 2016

### Staff: Mentor

The easiest way to prove something (there are always many options) depends only on what you want to prove, not on the way it is written down.

7. Nov 11, 2016

### Mr Davis 97

I agree with your statement, but just as an example of where I might have the wrong impression is with "Book of Proof," whose table of contents are here.

As you can see, it has one section "How to Prove Conditional Statements," and another "Proving Non-Conditional Statements," so it makes it seem as though proofs can be categorized by how they are written. Is the author wrong to organize the book in this way?

8. Nov 12, 2016

### Staff: Mentor

This is obviously a book about proofs. Therefore the author needed to partition it into chapters.

Most proofs are build by a combination of several techniques. E.g. $A \Rightarrow B \Rightarrow C$ may be shown by
"Given $A$, we assume $\lnot B$ which leads to a contradiction, next we can directly conclude $B \Rightarrow C$"
which is a frequently used pattern. There are proofs that fill entire books. Can you imagine they were done by a single technique?

What the author calls "Non-Conditional Statements" starts with "if-and-only-if" (iff) statements, i.e. equivalences. But these are simply two conditional statements combined:
"$A \Longleftrightarrow B\;$" is identical to "$A \Longrightarrow B \wedge B \Longrightarrow A\;$".
By the way, this does not mean that either were true (and therefore both were true). They both maybe false:
"$5$ is divisible by $2$, if and only if $3$ is divisible by $2$" is a true statement although neither $5$ nor $3$ is actually divisible by $2$.