What are the steps of proof?

Suppose you need to solve a crime mystery. You survey the crime scene, gather the facts, and write them down in your memo pad. To solve the crime, you take the known facts and, step by step, show who committed the crime. You conscientiously provide supporting evidence for each statement you make.

Amazingly, this is the same process you use to solve a proof. The following five steps will take you through the whole shebang.

1. Get or create the statement of the theorem.

The statement is what needs to be proved in the proof itself. Sometimes this statement may not be on the page. That's normal, so don't fret if it's not included. If it's missing in action, you can create it by changing the geometric shorthand of the information provided into a statement that represents the situation.

2. State the given.

The given is the hypothesis and contains all the facts that are provided. The given is the what. What info have you been provided with to solve this proof? The given is generally written in geometric shorthand in an area above the proof.

3. Get or create a drawing that represents the given.

They say a picture is worth a thousand words. You don't exactly need a thousand words, but you do need a good picture. When you come across a geometric proof, if the artwork isn't provided, you're going to have to provide your own. Look at all the information that's provided and draw a figure. Make it large enough that it's easy on the eyes and that it allows you to put in all the detailed information. Be sure to label all the points with the appropriate letters. If lines are parallel, or if angles are congruent, include those markings, too.

4. State what you're going to prove.

The last line in the statements column of each proof matches the prove statement. The prove is where you state what you're trying to demonstrate as being true. Like the given, the prove statement is also written in geometric shorthand in an area above the proof. It references parts in your figure, so be sure to include the info from the prove statement in your figure.

5. Provide the proof itself.

The proof is a series of logically deduced statements — a step-by-step list that takes you from the given; through definitions, postulates, and previously proven theorems; to the prove statement.

Remember the following:

• The given is not necessarily the first information you put into a proof. The given info goes wherever it makes the most sense. That is, it may also make sense to put it into the proof in an order other than the first successive steps of the proof.

• The proof itself looks like a big letter T. Think T for theorem because that's what you're about to prove. The T makes two columns. You put a Statements label over the left column and a Reasons label over the right column.

• Think of proofs like a game. The object of the proof game is to have all the statements in your chain linked so that one fact leads to another until you reach the prove statement. However, before you start playing the proof game, you should survey the playing field (your figure), look over the given and the prove parts, and develop a plan on how to win the game. Once you lay down your strategy, you can proceed statement by statement, carefully documenting your every move in successively numbered steps. Statements made on the left are numbered and correspond to similarly numbered reasons on the right. All statements you make must refer back to your figure and finally end with the prove statement. The last line under the Statements column should be exactly what you wanted to prove.

In addition to such techniques as direct proof, proof by contraposition, proof by contradiction, and proof by cases, there is a fifth technique that is quite useful in proving quantified statements: Proof by Induction!

Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)

What Is Proof By Induction

Inductive proofs are similar to direct proofs in which every step must be justified, but they utilize a special three step process and employ their own special vocabulary.

Inductive Process

Steps for proof by induction:

1. The Basis Step.
2. The Hypothesis Step.
3. And The Inductive Step.

Where our basis step is to validate our statement by proving it is true when n equals 1. Then we assume the statement is correct for n = k, and we want to show that it is also proper for when n = k+1.

The idea behind inductive proofs is this: imagine there is an infinite staircase, and you want to know whether or not you can climb and reach every step.

Staircase Analogy

If you can reach the first step (basis step), you can get the next step. And if you can ascend to the following step, then you can go to the one after it, and so on. Therefore, if it is true for the first step, then we will assume it is also appropriate for the kth step (guess). What is more, if it is correct for the kth step, it must be proper for the k+1 step (inductive).

So, the idea behind the principle of mathematical induction, sometimes referred to as the principle of induction or proof by induction, is to show a logical progression of justifiable steps.

Sometimes it’s best to walk through an example to see this proof method in action.

Example #1

Induction Proof Example — Series

That’s it!

We write our basis step, declare our hypothesis, and prove our inductive step by substituting our “guess” when algebraically appropriate.

Now, I do want to point out that some textbooks and instructors combine the second and third steps together and state that proof by induction only has two steps:

1. Basis Step.
2. Inductive Step.

While this is perfectly fine and reasonable, you must state your hypothesis at some point at the beginning of your proof because this process is only valid if you successfully utilize your premise. In addition, Stanford college has a handy PDF guide covering some additional caveats.

This means that you have first to assume something is true (i.e., state an assumption) before proving that the term that follows after it is also accurate.

I like to think of it this way — you can only use it if you first assume it!

While most inductive proofs are pretty straightforward there are times when the logical progression of steps isn’t always obvious. Therefore, we will have to be a bit creative.

For instance, let’s work through an example utilizing an inequality statement as seen below where we’re going to have to be a little inventive in order to use our inductive hypothesis.

Example #2

Inductive Proof Example — Inequality

Did you spot our sneaky maneuver?

By saying that (K+1) < (K+K) we were able to employ our inductive hypothesis and nicely verify our "k+1" step! Together we will look at numerous questions in detail, increasing the level of difficulty, and seeing how to masterfully wield the power of prove by mathematical induction.

Video Tutorial w/ Full Lesson & Detailed Examples

1 hr 48 min

• Introduction to Video: Proof by Induction
• 00:00:57 What is the principle of induction? Using the inductive method (Example #1)
• Exclusive Content for Members Only

• 00:14:41 Justify with induction (Examples #2-3)
• 00:22:28 Verify the inequality using mathematical induction (Examples #4-5)
• 00:26:44 Show divisibility and summation are true by principle of induction (Examples #6-7)
• 00:30:07 Validate statements with factorials and multiples are appropriate with induction (Examples #8-9)
• 00:33:01 Use the principle of mathematical induction to prove the inequality (Example #10)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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