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This is a pretty strange question, perhaps.

I've studied logic and know very well the difference between A --> B and B --> A.

However, there's a specific problem that I often encounter in proofs in maths that I find strange.

For example: you get a certain P.D.E. and want to prove that there exists a solution and that it is unique. Then you prove: "if f(x) is a solution, then f(x) must be of a certain form (for example, f(x) = e

^{x})". However, you also need to show that f(x) is a solution.

What I can't really imagine, is a situation in which I prove that if f(x) is a solution, then f(x) = e

^{x}, but then to find out that e

^{x}isn't a solution.

The only examples I could think of are "empty cases". For example, if you get the equation f'(x) +f(x) + 1 = f'(x) +f(x) + 2, I could say: if f(x) is a solution then f(x) = e

^{x}. Since f(x) is never a solution (cause there is no solution) we get a claim of the form F --> T/F, which is always "T".

But are there other examples, other than empty cases, in which a solution must have the form f(x), but f(x) isn't a solution?

Confused :-)