Example A quadratic equation over the complex numbers always has two roots.
The equation
In contrast, the equation:
In other words, the first equation has distinct roots, while the second does not.
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.
I don't know what to say here, but there are issues. I can mention Leibniz's law[?] (which we don't have an article on); this is the claim that distinct things can always be proved distinct as in the last section by some property. These ideas could be discussed under Identity. There is some discussion at Identity and change.
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