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 "Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle "<math> |x| > 1 } " and "$x\geq 7$" </math>", etc.

Resoucres

• Proof by Cases, Millersville University