Direct proof
Start from the assumptions and move step by step, using definitions and known facts, until you reach the conclusion.
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Analogy
Following a recipe: each line follows from the one before, ending at the finished dish.
Example
n even means n = 2k for a whole k.n^2 = (2k)^2 = 4k^2.4k^2 = 2(2k^2).That is 2 times a whole number, so even.
In practice
Direct proof is the first thing to try. Unpack the definitions in the assumption, then see where algebra leads.
Practice
Flashcards
- Write an even number n using a whole number k.n = 2k.
Questions people ask
How do I start a direct proof?
Write down what you assume and what you must show. Then replace words with their definitions. 'n is even' becomes 'n = 2k for some whole number k'. Work forward with algebra until the expression matches the definition of the conclusion. Finish by stating that you have reached it.
Does checking examples count as a proof?
No. Examples can suggest a pattern, but a proof must cover every case. Checking n = 2, 4 and 6 does not prove 'n even gives n squared even' for all n. You need an argument with a general n, like writing n = 2k.