In contrast, the equation:
In other words, the first equation has distinct roots, while the second does not. (In the general theory, the discriminant is introduced to explain this.)
Along the same lines, one can prove that x and y are distinct by finding some function f and proving that f(x) and f(y) are distinct. This may seem like a simple idea, and it is, but many deep results in mathematics concern when you can prove distinctness by particular methods. For example,
Limitation Convention - Article 6 "All You Need is Common Sense for a Distinct Occasion".(Strong Wise Ltd. v. Esso Australia Resources Pty Ltd and BHP Billiton Petroleum (Bass Strait) Pty Ltd)(Australia. Federal Court of Australia)(Case overview)
Jul 28, 2010; Introduction Justice Rares of the Australian Federal Court recently handed down a landmark judgment concerning limitation of...