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Casework is basically what you would expect: considering all the possible cases of something and proving the given statement for all of them. This technique is especially powerful in combinatorics, but it works in all parts of math.
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A detective hard at work.
A good detective tries to find all possible scenarios that could've happened at the scene. The detective then tests each scenario to see whether it is plausible, and then tries to prove it with evidence and material from the scene.
Casework is very much like this process that detectives go through. As mathematicians, we try to prove statements or solve problems. The evidence and material we have are what we call in the math world theorems, postulates, definitions, etc. Each scenario is called a case, and the work we do to prove/solve each case is called casework.
In short, casework is a rigorous type of proof/work that proves the statement to be true/finds the solution of a problem in all possible cases/scenarios/circumstances.
Let's say that your math teacher wrote a statement on the board. It reads:
All perfect squares are positive or zero.
Your teacher then asks you to prove it. There are many ways in proving this, but we will use casework.
Here's the basic breakdown:
You know that perfect squares can be written in the form , where is a real number?
And that this can be positive, negative, or zero?
Now, it's time to do some casework for each possible case (when the number is positive, zero, or negative).
Case 1: when is positive
If is positive, then must be positive because:
We've got our first case down, now we have two more to go!
Case 2: when is zero
If is zero, then must be zero because:
Two down, one to go!
Case 3: when is negative
If is negative, then must be positive because:
Great; now that we have proved all possible cases and showed that is positive or zero for each of these cases, we can conclude that the statement given by our teacher is indeed true.
Casework's potential is not limited to this proof; it is widely used for proving theorems in combinatorics, solving problems involving probability, and more!
Try to use casework when you are doing math when applicable, and continue on with your journey at expii.
