So what’s the difference between proof by cases and proof by exhaustion?
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
Actually, they’re the same thing.
And that’s exactly what you’re going to learn about in today’s discrete math class.
Let’s go!
But first let’s discuss some basic terminology.
A mathematical statement that aims to exhaust all possibilities by splitting the problem into parts and considering each piece or case separately is called proof by cases, sometimes referred to as proof by exhaustion.
Definition
The idea behind the proving method is that we break the proof into smaller, more manageable conditions and prove that the overall claim holds for every case.
Exhaustion Proof — Definition
How Does Proof By Cases Work
Proof by exhaustion must show that a statement is true for each unavoidable configuration as noted by Grand Valley State University. Therefore, we must wisely choose how we split our problem into parts and ensure that we consider every possible case. Otherwise, the proof is futile.
Sometimes all we will need to do is show that the argument holds for a number is odd or even. Other times we may need to show that a claim holds for a range of numbers or inequalities.
So, how many cases will we need?
Typically, we will find that most proofs of this type will contain two to four cases, but there isn’t a limit to the number of cases you can have. For example, the famous “Four-Color Map Theorem” has over 600 cases! Don’t worry — we will focus on questions with four cases or less!
There are times when you will start a proof by clearly stating each possible case and then showing each case is true using clear and logical steps. Other times, you may begin by using direct or indirect proofs but will pivot to using proof by cases to complete your reasoning. We will look at both scenarios within this lesson.
Example #1
But for now, let’s look at a few examples of proof by cases.
Proof By Cases — Example
Notice how this claim is structured in such a way that leads you to the notion of splitting up the problem into two parts: either n > 1 or n < -1. Here's a big hint: whenever we have absolute value - use proof by cases!
Example #2
Now let’s look at another example.
Proof By Exhaustion — Claim
Case 1
Case 2
Case 3
Not bad!
As with all proofs, we are trying to convince the audience of our claim’s validity; therefore, we must be as clear and concise as possible.
Together we will work through numerous examples to ensure the method of exhaustion is mastered.
Video Tutorial w/ Full Lesson & Detailed Examples
1 hr 44 min
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