Mathematical Writing - Vivaldi Franco 2014
Proof by Cases
Forms of Argument
Some mathematical arguments are made tidier by breaking them into a number of cases, of which precisely one holds, while all lead to the desired conclusion.
EXAMPLE. Consider the following statement:
The solution set of the inequality is the complement of the open interval .
PROOF. Let be a real number. There are three cases.
(1)
(2)
(3)
So the required solution set is the collection of the values of such that , or , or , which is the union of two rays:
This set is obtained from the real line by removing the open interval , as claimed.
The opening sentence ’Let be a real number’ acknowledges that all real values of will have to be considered. The second sentence announces that the proof will branch into three cases, determined by the sign of the expressions within absolute value. Note the careful distinction between strict and non-strict inequalities, to avoid missed or repeated values of .
The structure of this proof is shaped by the structure of the absolute value function, which is piecewise-defined—see Sect. 5.6. The presence of piecewise-defined functions usually leads to proofs by cases.
EXAMPLE. A proof by cases on divisibility.
Let be an integer; then is divisible by 30.
PROOF. We factor the integer 30 and the polynomial :
For each prime we will show that, for any integer , at least one factor of is divisible by .
1.
2.
3.
The proof is complete.
After the factorisations we declare our intentions. There are three cases, one for each prime divisor of 30. Each prime in turn leads to cases, corresponding to the possible values of the remainder of division by . To simplify book-keeping, we consider also negative remainders, and then pair together the remainders which differ by a sign. Note the presence of the symbols and within the same sentence. The positive sign in the first expression matches the negative sign in the second expression, and vice-versa.