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I'm trying to understand the basis of contradiction and I feel like I have understood the ground rules of it.

For example: Show that the square of an even number is an even number using a contradiction proof.

What I have is: Let n represent the number.

n is odd if n = 2k + 1, where k is any number

n is even if n = 2k, where k is any number

We must prove that if n^2 is even, then n is even.

How do I proceed on from here?

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  • $\begingroup$ You have already posted a question about this not too long ago. $\endgroup$ – fkraiem Feb 10 '14 at 18:52
  • $\begingroup$ Hint: directly plug in $2k+1$ for $n$ then see that you receive a contradiction. You want to assume $n^2$ is odd for contradiction. $\endgroup$ – user59999 Feb 10 '14 at 18:57
  • $\begingroup$ @Sentrl,you are welcome,please look on my answer,you simple should assume contradiction and you will see by this step how to apply this emthod $\endgroup$ – dato datuashvili Feb 10 '14 at 19:08
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We prove the contrapositive. In this case, we want to prove

$n^2$ even implies $n$ even

which is equivalent to the contrapositive

$n$ not even implies $n^2$ not even

or in other words

$n$ odd implies $n^2$ odd.

If $n$ is odd, then $n=2k+1$ then \begin{align*} n^2 &= (2k+1)^2 & \text{substituting in } n=2k+1 \\ &= 4k^2+4k+1 & \text{expanding} \\ &= 2(2k^2+2k)+1 \end{align*} which is odd, since it has the form $2M+1$.

We can essentially turn this into a proof by contradiction by beginning with "If $n$ is odd and $n^2$ is even...", then writing "...giving a contradiction" at the end. Although this should be regarded as unnecessary.


The other direction, i.e.,

$n$ even implies $n^2$ even

can also be shown in a similar way: If $n=2k$, then $n^2=(2k)^2=4k^2=2(2k^2)$ which is even.

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If you insist by contradiction...then consider some $n$ that is even, then: $$n = 2k$$ Where $k$ is some natural number not $0$. Assume that $n^2$ is not even, but then contradicting the fact that $n^2 = (2k)^2 = 4k^2 = 2(2k^2)$ is even.

Alternatively, if $n^2$ is even but $n$ is odd, then $n = 2k+1$ so $$(2k+1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1$$ ...contradicting the fact that $n^2$ is assumed to be even.

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