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Let $M$ be a set and let d be a function from $M \times M$ into $\mathbb R$ which satisfies the properties:

  1. $d(x,y) = 0$ if and only if $x=y$
  2. $d(x,y) = d(y,x)$ for all $x,y \in M$
  3. $d(x,z) \leq d(x,y) + d(y,z)$ for all $x,y,z \in M$

Prove that $d: M \times M \to [0, \infty)$.

MPW
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C.Sherman
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    I presume that property (3) is some variety of the triangle inequality, but you posted the question without completing the statement of property (3). Please clarify. – MPW Dec 01 '16 at 16:00
  • The (3) property is the triangle inequality. My apologies. I am new to the site so still working out formatting issues. The question is complete now. – C.Sherman Dec 01 '16 at 16:10
  • Hint: for an arbitrary pair $x,y\in M$, see what you can yield from looking at $d(x,x)$ or $d(y,y)$. – Meitar Dec 01 '16 at 16:32
  • So you have axioms of metric but you have omitted the condition that it is non-negative. And you are trying to show that non-negativity follows from other conditions? You should probably clarify in the question whether this is what you are after. If yes, this can be found in the Wikipedia article on metric spaces (link to the current revision.) See also this question. – Martin Sleziak Dec 01 '16 at 16:33
  • You may also be interested in my answer to this question which shows that you can actually reduce the required properties to only two. – MPW Dec 01 '16 at 20:05

2 Answers2

1

Proceed by contradiction:

suppose there exists a pair $(x, y) \in M \times M$ such that $d(x, y) < 0$, use symmetry and triangle rule with $z = x$ and see what happens!

th0masb
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Just take $z=x$ to obtain $$0 = d(x,x) \leq d(x,y) + d(y,x) = 2d(x,y)$$ so that $d(x,y)\geq 0$ as required.

The first "$=$" is from property 1, the "$\leq$" is from property 3, and the second "$=$" is from property 2.

MPW
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