5 Proof by counter example

Proof by counter example is only used to show that a proposition is not true. A ``counter example'' is an actual case that contradicts a proposition. Note that proof by counter example is only useful to prove a proposition wrong. On the other hand, proof by contradiction is often used to prove a proposition right.

We can change proposition 1 a little bit to illustrate this:

$$\forall i \in I \wedge i \ge 2, \forall j \in I \wedge 1 \l... ... \times k) \rightarrow ((j < \sqrt{i}) \vee (k < \sqrt{i}))$$ (3)

All I did was to change the less-than-or-equal-to into less-than. Proposition 3 is false because I can easily find a counter example: $i = 9, j = 3, k = 3$. This counter example essentially confirms the following:

$$\exists i \in I \wedge i \ge 2, \exists j \in I \wedge 1 \l... ...= j \times k) \wedge ((j = \sqrt{i}) \wedge (k = \sqrt{i}))$$ (4)

Note that proposition 4 is not proposition 1! Although the predicate is the same, the quantifiers are different. Proposition 1 implies proposition 4, but not the other way around. In other words, proposition 1 is more general than proposition 4.

Even though our counter example confirms proposition 4, proposition 4 is too specific to be generally useful.