Metric - Revision history

InfernalAtom683 at 14:01, 29 June 2026

2026-06-29T14:01:40Z

InfernalAtom683 at 16:22, 12 June 2026

2026-06-12T16:22:39Z

← Older revision		Revision as of 00:22, 13 June 2026
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	&0\le d(x,y)		&0\le d(x,y)
	\end{align}</math>}}		\end{align}</math>}}

			{{Property\|title=Reverse Triangle Inequality\|property=Suppose that <math>(X,d)</math> is a metric space and <math>x,y,z\in X</math>. Then <math display="block">\|d(x,y)-d(x,z)\|\le d(y,z).</math>}}
			{{Proof\|proof=By triangle inequality,
			<math display="block">d(x,y)\le d(x,z)+d(z,y).</math>
			By symmetry, re-arranging the inequality yields
			<math display="block">d(x,y)-d(x,z)\le d(y,z).</math>

			Utilizing the triangle inequality again,
			<math display="block">d(x,z)\le d(x,y)+d(y,z).</math>

InfernalAtom683: /* Definitions */

2026-06-12T12:03:29Z

Definitions

← Older revision		Revision as of 20:03, 12 June 2026
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	== Definitions ==		== Definitions ==
	Let <math>X</math> be a given non-empty set. A ~~metirx~~ on <math>X</math> is a function <math display="block">d: X\times X\to \mathbb{R}</math>,		Let <math>X</math> be a given non-empty set. A metric on <math>X</math> is a function <math display="block">d: X\times X\to \mathbb{R},</math>

	that satisfies the following four properties for all <math>x,y,z\in X</math>:		that satisfies the following four properties for all <math>x,y,z\in X</math>:

InfernalAtom683: /* Properties */

2026-06-12T12:02:59Z

Properties

← Older revision		Revision as of 20:02, 12 June 2026
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	By symmetry, <math>d(x,y)=d(y,x)</math>. Thus		By symmetry, <math>d(x,y)=d(y,x)</math>. Thus
	<math display="block">\begin{align}		<math display="block">\begin{align}
	0&\le&d(x,y)+d(x,y)\\		&0\le d(x,y)+d(x,y)\\
	0&\le&2d(x,y)\\		&0\le 2d(x,y)\\
	0&\le&d(x,y)		&0\le d(x,y)
	\end{align}</math>}}		\end{align}</math>}}

InfernalAtom683: /* Properties */

2026-06-12T12:02:05Z

Properties

← Older revision		Revision as of 20:02, 12 June 2026
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	By symmetry, <math>d(x,y)=d(y,x)</math>. Thus		By symmetry, <math>d(x,y)=d(y,x)</math>. Thus
	<math display="block">\begin{align}		<math display="block">\begin{align}
	0&\le d(x,y)+d(x,y)\\		0&\le&d(x,y)+d(x,y)\\
	0&\le 2d(x,y)\\		0&\le&2d(x,y)\\
	0&\le d(x,y)		0&\le&d(x,y)
	\end{align}</math>}}		\end{align}</math>}}

InfernalAtom683: Created page with "In mathematics, a '''metric''' or '''distance function''' is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric sp..."

2026-06-12T11:39:26Z

Created page with "In mathematics, a '''metric''' or '''distance function''' is a function that defines a rigorous concept of distance between each pair of elements in a given set. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as open and closed sets, continuity, convergence, and limits in highly abstract spaces. A set equipped with a metric is called a metric sp..."

New page

In [[mathematics]], a '''metric''' or '''distance function''' is a [[function]] that defines a rigorous concept of distance between each pair of elements in a given [[set]]. By generalizing the physical concept of distance, metrics allow mathematicians to define topological concepts such as [[Open set|open]] and [[Closed set|closed sets]], [[continuity]], [[convergence]], and [[Limit|limits]] in highly abstract spaces. A set equipped with a metric is called a [[metric space]].

== Definitions ==
Let <math>X</math> be a given non-empty set. A metirx on <math>X</math> is a function <math display="block">d: X\times X\to \mathbb{R}</math>,

that satisfies the following four properties for all <math>x,y,z\in X</math>:

# '''Non-negativity:''' <math>d(x,y)\ge 0</math>
# '''Identity of indiscernibles:''' <math>d(x,y) = 0 \Longleftrightarrow x=y</math>
# '''Symmetry:''' <math>d(x,y)=d(y,x)</math>
# '''Triangle inequality:''' <math>d(x,z)\le d(x,y)+d(y,z)</math>.

== Properties ==
{{Property|property=Non-negativity is implied by the other three properties.}}{{Proof|proof=Let <math>x,y\in X</math>. By triangle inequality,
<math display="block">d(x,x)\le d(x,y)+d(y,x).</math>
By the identity of indisscernbles, <math>d(x,x)=0</math>. Thus
<math display="block">0\le d(x,y)+d(y,x).</math>
By symmetry, <math>d(x,y)=d(y,x)</math>. Thus
<math display="block">\begin{align}
0&\le d(x,y)+d(x,y)\\
0&\le 2d(x,y)\\
0&\le d(x,y)
\end{align}</math>}}