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I have this problem:

Let the following operation be defined in $N_0$:

$$ \star : (x,y) \in N_0 \times N_0 \to x^2 + y^2 \in N_0 $$

Define if: the structure has a neutral elemnt.

I think it doesn't, because, from the definition of netrual element:

$$ e*x = x*e = x $$

In this case that never happens:

$0 \star y = y^2$

$1 \star y = 1+y^2$

$2 \star y = 4+y^2$

And so on... So we never get just $y$, but at the very least we get $y^2$.

How can I formalize this? I was thinking about induction, but the assert it's false for any possible base, so in the induction step you can't use the assert.

2 Answers 2

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Assume there is a neutral element. Then

$x*y=x^2+y^2=y$ for all $y\in\mathbb{N}_0$

implies

$x^2=1-1^2=0=2-2^2=-2$

a contradiction. Whence there is no neutral element.

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    Would this line reasoning work for any similar problem about the existence of the neutral element? – 2017-01-25
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Consider that, for a neutral element $e$, we need that $e \star 2 = e^2+2^2 = 2$. Thus $e^2 = -2$, which is not possible with $e\in \Bbb N_0$

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    Yeah that's obvious I'm ashamed I didn't think of it – 2017-01-25
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    Sometimes the "easy" proofs are hard to see. – 2017-01-25