What is your favourite counterexample from the world of maths?

Answer 1

in xy plane,
slope of x-axis=0,
slope of y-axis=infinity,
but product of these two is not "-1".
so,
something is there in infinity.

Answer 2

The answer about connected subsets isn't right. The intersection of an ascending chain of subsets is just the smallest subset (i.e. first subset in the chain), which is connected. For that matter, the union of this chain is also connected (think of what it would mean if there was a separation of this union; where must the first subset of the chain lie? the second?) .

As for a different counterexample, the equation 3x^3+4y^3+5z^3=0 has no solutions with all of x, y, and z non-zero integers. (This is actually difficult to show.) This a counterexample for what? It turns out that this equation does have solutions if you allow p-adic integers. There is a famous principle, the so-called Hasse Principle, that says if an equation has solutions in all of the p-adic fields and also the real numbers, then there is a solution over the rational numbers. (For the above equation, if there is a rational solution, then clear the denominators to get a solution over the integers.) The above equation shows that this principle doesn't always hold. (Yes, there are places where it does hold. Quadratic forms, for example.)

Answer 3

Suppose that you have an increasing (by containment) series of connected subsets. Then the intersection of this set may not be connected. At first glance most people would feel that the intersection would also be connected. There are many counterexamples in the plane which are quite interesting.

Whoops, I meant decreasing sequence on connected sets.

Answer 4

That an infinite number of 2's may be factored from any 'rational' representation (a/b) of √2 thus countering the assumption that (a/b) is reduced to it's lowest terms.


Doug