Proving Distinctness
In order to prove that two things x and y are distinct, it often helps to find some property that one has but not the other. For a simple example, if for some reason we had any doubt that the roots 1 and 2 in the above example were distinct, then we might prove this by noting that 1 is an odd number while 2 is even. This would prove that 1 and 2 are distinct.
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,
- The Hahn–Banach theorem says (among other things) that distinct elements of a Banach space can be proved to be distinct using only linear functionals.
- In category theory, if f is a functor between categories C and D, then f always maps isomorphic objects to isomorphic objects. Thus, one way to show two objects of C are distinct (up to isomorphism) is to show that their images under f are distinct (i.e. not isomorphic).
Read more about this topic: Distinct, In Mathematics
Famous quotes containing the word proving:
“The momentary charge at Balaklava, in obedience to a blundering command, proving what a perfect machine the soldier is, has, properly enough, been celebrated by a poet laureate; but the steady, and for the most part successful, charge of this man, for some years, against the legions of Slavery, in obedience to an infinitely higher command, is as much more memorable than that as an intelligent and conscientious man is superior to a machine. Do you think that that will go unsung?”
—Henry David Thoreau (1817–1862)