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Why does $d(x,y)=|x^3-y^3|$ define a metric space but $d(x,y)=|x^2-y^2|$ does not on real numbers?

I know how to show $d(x,y)=|x^3-y^3|$ is a metric space using the axioms, but I thought that property (M1) was also satisfied for $|x^2-y^2|$, but in my answers it says because $d(−1, 1) = 0$ but $−1 ≠ 1$?

But if I take two real numbers say $x=2$ and $x=3$ for the first metric, I also don't get $x$ equal to $y$? So I am rather confused at this notion! If someone could clarify this for me that would be great, thank you!

Bernard
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12589user
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    One of the axioms requires that $d(x,y)=0\iff x=y$. – lulu Nov 27 '19 at 14:42
  • yes, but if i solve this |x^2-y^2|=0<=>x^2=y^2<=>x=y – 12589user Nov 27 '19 at 14:44
  • No. You have already given $x=-1, y=1$ as a counterexample to that. $x^2=y^2\implies x=\pm y$. – lulu Nov 27 '19 at 14:45
  • Yes that was in my solutions, but I don't understand how we know to take these two numbers as counterexamples. If we take two different Real numbers for x^3 - y^3 we could argue the same point could we not? – 12589user Nov 27 '19 at 14:47
  • Just try it. You'll need to produce two distinct real numbers with the same cube. – lulu Nov 27 '19 at 14:47
  • Saying that one axiom is not always satisfied does not mean it is never satisfied. – Bernard Nov 27 '19 at 14:58

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Intuitively, the property in question says the distance between any pair of different points is nonzero. For the proposed metric $d(x,y)=|x^2-y^2|$ that is not true because $d(-1,1)=|(-1)^2-1^2|=|1-1|=0$. We have found two different points at distance zero, violating the property. Having found one example that violates the axiom is sufficient to show this is not a metric.

Your first comment is incorrect. The first equivalence, $|x^2-y^2|=0 \iff x^2 = y^2$ is true, but $x^2=y^2 \iff x=y$ is not. Take $x=-1,y=1$ and the left is true but the right is false.

Ross Millikan
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  • Ok, this has helped very much thank you! I am new to metric spaces and just want to make sure I have everything correct thank you :-) – 12589user Nov 28 '19 at 10:59