[SOLVED] Proving a theorem without using contradiction

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Nov 16th 2009, 03:41 PM

Plato

Quote:

Originally Posted by racewithferrari

The following theorem has the form of an implication.
If n is an integer and 3n+2 is even, then n is even.

Suppose that $\displaystyle 3n+2=2k$ then $\displaystyle 3n=2k-2$.
What does that tell you?

Nov 16th 2009, 03:42 PM

Drexel28

Quote:

Originally Posted by racewithferrari

The following theorem has the form of an implication.

If n is an integer and 3n+2 is even, then n is even.

Give a direct proof of this theorem without using contradiction.

Since $\displaystyle 3n+2$ is even we may state WLOG that $\displaystyle 3n+2=2z\implies 3n=2(z-1)$ and since $\displaystyle 2\nmid 3$ it must be true that $\displaystyle 2\mid n$. The conclusion follows.