Proving Theorems Not in "If-Then" Form

[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?

 

[mfb]

"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.

 

[fresh_42]

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.

 

[Mr Davis 97]

"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.

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

 

[Hercuflea]

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

 

[mfb]

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

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.

 

[Mr Davis 97]

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.

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.

• Part I: Fundamentals
  
  1. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Sets.pdf
  
  2. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Logic.pdf
  
  3. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Counting.pdf
• Part II: How to Prove Conditional Statements
  
  4. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Direct.pdf
  
  5. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Contrapositive.pdf
  
  6. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Contradict.pdf
• Part III: More on Proof
  
  7. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Other.pdf
  
  
  8. http://www.people.vcu.edu/%7Erhammack/BookOfProof/SetProofs.pdf
  
  9. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Disproof.pdf
  
  10. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Induction.pdf
• Part IV: Relations, Functions and Cardinality
  
  11. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Relations.pdf
  
  12. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Functions.pdf
  
  13. http://www.people.vcu.edu/%7Erhammack/BookOfProof/Cardinality.pdf

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?

 

[fresh_42]

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?

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 $2$.