There is this question, but the definition of "even" and "odd" that I am using uses integers instead of just natural numbers; i.e.,

  • An integer $n$ is even iff there is some integer $k$ such that $n=2k$.
  • An integer $n$ is odd iff there is some integer $k$ such that $n=2k+1$.

Here is what I have so far:

First we show that an integer $n$ is even or odd. We first use induction on the positive integers. For the base case, $1=2\cdot0+1$ so we are done. Now suppose inductively that $n$ is even or odd. If $n$ is even, then $n=2k$ for some $k$ so that $n+1=2k+1$ (odd). If $n$ is odd, then $n=2k+1$ for some $k$ so that $n+1=2(k+1)$ (even). This closes the induction, so every $n\in\mathbf{Z}^+$ is even or odd.

Now we show every $n\in\mathbf{Z}^-$ is even or odd. Let $n\in\mathbf{Z}^-$. Then $n=-k$ for some $k\in\mathbf{Z}$ (I think this follows immediately from most definitions of the integers.). Suppose $k$ is even. Then $k=2j$ for some $j$ so that $n=-k=-2j=2(-j)$ (even). Now suppose $k$ is odd. Then $k=2j+1$ for some $j$ so that $n=-k=-(2j+1)=-2j-1=-2j-1+1-1=-2j-2+1=2(-j-1)+1$ (odd).

For $0$, note that $0=2\cdot0$ (even).

Now we show that $n\in\mathbf{Z}$ cannot be both even and odd. Suppose for the sake of contradiction that $n\in\mathbf{Z}$ is both even and odd. Then there are integers $k,j$ such that $n=2k=2j+1$. This implies that $2(k-j)=1$ (like in the referenced question). So we must show that $1$ cannot be even in order to complete the proof.

This is where I am having trouble. I know that if I let $f:\mathbf{Z}\to\mathbf{Z};x\mapsto2x$ be a function, then $f$ is increasing so since $f(0)=0$ and $f(1)=2$ and $0<1<2$, there is no integer $m$ such that $f(m)=1$. But this seems complicated so I was wondering if there was an easier way to do this.

So my real question is: how can I show that $1$ is not even?

(This is not homework.)

  • $\begingroup$ $f(m)=1 \implies 2m=1 \implies 2=\frac {1}{m} $ which is not a integer for any $m \in \mathbb Z $ $\endgroup$ – Theorem Jul 4 '12 at 20:57
  • $\begingroup$ @russell11, I think you meant a function $f:\mathbb{Z}^+ \to \mathbb{Z}^+$ (or whatever the appropriate notation is for 'nonnegative'). Otherwise you and Theorem converge on the right idea; suppose $1=2m$ for some integer $m$. $m>0$ comes immediately, as twice a non-positive integer is non-positive. $m<1$ comes from the function being increasing. $\endgroup$ – Eugene Shvarts Jul 4 '12 at 21:02

To show that $1$ is not even:

I assume you can prove or accept that $a \cdot 0 = 0$ for all $a \in \mathbb{Z}$, the product of a positive and negative number is negative, and $2a > a$ when $a >0$.

If $1$ is even, then there must exists $a < 1$ such that $2a = 1$. However, the only $a < 1$, which is an integer, is $0$ and clearly $2 \cdot 0 = 0$. So $1$ can not be be written as $2 \cdot a$ for any $a \in \mathbb{Z}$. $1$ is odd.

  • 1
    $\begingroup$ Do you want to say, "So $1$ cannot be written as $2\cdot a$ for any $a\in\mathbb{Z}$" in the end? $\endgroup$ – Paul Jul 4 '12 at 21:13
  • $\begingroup$ @Paul Yes. Added the missing "not". $\endgroup$ – William Jul 4 '12 at 21:21

Hint $\ $ Your induction step uses $\rm\:n\,$ even $\rm\,\Rightarrow\: n\!+\!1\:$ odd, and $\rm\:n\,$ odd $\rm\,\Rightarrow\:n\!+\!1\:$ even. The converses are both true, e.g. $\rm\,n\!+\!1\,$ odd $\,\Rightarrow$ $\rm\,n\!+\!1 = 2k\!+\!1\,$ $\Rightarrow$ $\rm\,n = 2k,\,$ since $\rm\,j\!+\!1 = k\!+\!1\,$ $\Rightarrow$ $\rm\,j = k\,$ by Peano axioms. Thus if $\rm\:n\!+\!1\,$ is both even and odd then so too is $\rm\,n.\,$ So, contrapositively, in your induction step you can lift "$\rm n\:$ is not both even and odd" from $\rm\,n\,$ up to $\rm\,n\!+\!1\,$ and hence prove by induction that every natural is even or odd, but not both (this is essentially my hint in the linked answer).

  • $\begingroup$ Thanks, this is really neat. I didn't understand your hint in the other question, but now I do. $\endgroup$ – russell11 Jul 5 '12 at 1:18

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