MATH.6:4.2 - Work out what failure requires
Keep the assumptions true and negate the conclusion. Preserve the order and permitted dependence of choices. The following ordinary forms cover common starting points; P and R denote the stated properties, and each variable keeps its declared domain.
| Conclusion to defeat | Construction or argument needed for failure |
|---|---|
Every x has property P(x). | One allowed x for which P(x) fails. |
Some y has property P(y). | A reason why P(y) fails for every allowed y. |
For every x, some y satisfies R(x,y). | One allowed x for which every allowed y fails. |
Some y satisfies R(x,y) for every x. | For each allowed y, an x that defeats it; this x may depend on y. |
An equation between operations can often be defeated by one input tuple. Failure of transitivity requires three elements with the first two links present and the third absent. Start with those witnesses and use them to constrain the construction.
Where failure contains a universal requirement, give its argument or a complete finite case distinction. Finding one unsuccessful candidate for an existential conclusion leaves the other candidates open.