Proofs:Cases

Some proofs are easier to do if we split them up into two or more cases.

Example: Proof that $x^{2}\geq 0$ for all real numbers. We can break this up into three cases: $x^{2}\geq 0$ , $x=0$ , and $x<0$ . If $x>0$ , then $x^{2}=x*x\geq 0$ , since the product of two positive numbers is positive. If $x=0$ , then $x^{2}=0*0=0$ . If $x<0$ , then $x^{2}=x*x$ is the product of two negative numbers, which is positive. Thus, $x^{2}\geq 0$ for all three cases, and is therefore true for all real numbers x.

Examples of other ways to break sets into cases:

Integers: "z is negative" and "z is nonnegative", "z is even" and "z is odd", etc.
Real numbers: "x is rational" and "x is irrational", " $x<7$ " and " $x\geq 7$ ", " $|x|\leq 1$ " and " $|x|>1$ " and " $x\geq 7$ " </math>", etc.