Example
Consider the following theorem (which is a case of the Pigeonhole Principle):
If three objects are each painted either red or blue, then there must be two objects of the same color.
A proof:
Assume without loss of generality that the first object is red. If either of the other two objects is red, we are finished; if not, the other two objects must both be blue and we are still finished.
This works because exactly the same reasoning (with "red" and "blue" interchanged) could be applied if the alternative assumption were made, namely that the first object is blue.
Read more about this topic: Without Loss Of Generality
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)