Math 111

http://www.fresnostate.edu/csm/math/students/latex.html

We have four different techniques that we use to approach a problem. Those are:

1. Direct Proof

2. Proof by Contrapositive

3. Proof by Contradiction

4. Proof by Induction

In addition to those, we may also have equivalence statements, meaning that we have to prove both directions. It can sometimes be difficult to determine which technique to use on a given problem, so below are a few tips from our Math 111 tutors. 

What do you do first when you look at a problem?

• Identify what the problem is asking you to prove as well as what is given.

• Find relevant theorems, corollaries, and lemmas that might help you outline your proof.

• List out definitions to help you better understand how to work with different concepts.

How do you know when to use a direct proof?

• A direct proof is often the first method to try, unless another method seems more useful.

• If you run into any problems with a direct proof, often those problems help you identify which method to try next.

How do you know when to use a proof by contrapositive?

• If a statement assumes that something is not true, and you need to prove that something else is not true, then a contrapositive proof might be a good first option.

• Oftentimes, a proof by contradiction will be more convenient than a proof by contrapositive.

How do you know when to use a proof by contradiction?

• A proof by contradiction is useful when we want to prove that a condition does not hold.

• We can work with more definitions and theorems when we assume that a condition does hold.

• A proof by contradiction also allows us to work with what we want to show, something we can't do with any other proof technique.

How do you know when to use a proof by induction?

• Induction is usually used when we want to prove that a statement is true for every integer after some number.

• Proofs by induction follow a pattern, which makes our outline much easier to create.

• Your inductive hypothesis should show up somewhere in your proof. Make sure to keep an eye out for where you can use it.

What else do you recommend when trying to prove things?

• GIVE YOURSELF PLENTY OF TIME. When the assignment is first assigned, try to read through each problem and identify the ones you might find challenging as well as those that seem simpler. Try to knock out the easy ones first and give your brain time to think about the more difficult ones. If none of your strategies have worked, visit office hours or tutoring for a new perspective.

• Always do scratch work first to clearly map out your thoughts. This can serve as an outline to your actual proof.

• Don't assume that something is obvious or trivial unless told otherwise. Every statement must be supported by a theorem, definition, or something that has been shown to be true.

The only way to get better at writing proofs is to practice. Attached is a worksheet with problems using all four techniques. For each problem, identify which proof technique you would use, then try to prove it. Good luck!