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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
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5. d(a, c) ≤ d(a, b) + d(b, c)
5. d(a, c) ≤ d(a, b) + d(b, c)
If an interval in the interval class of a equals **s**[a] and an interval in the interval class of b equals **s**[b], then their product, reduced modulo the interval of equivalence **O** which is  **s**[**P**], will be **s**[i + j] mod **O**. Hence to get an interval in the class of **s**[i + j] mod **O** other than **s**[i + j] mod **O** as a product, either the interval in the class of **s**[i] must be other than **s**[i], or the interval in the class of **s**[j] must be other than **s**[j]. If always only one of the intervals is different than the defining interval for its class, then || **s**[i + j] mod **O** || equals ||**s**[i]|| + ||**s**[j]||. However, there may be overlap, so that the first interval is not in the class for **s**[i] and the second not in the class for **s**[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(**s**[i], **s**[j]) + d(**s**[j], **s**[k]) = || |i - j| || + || |j - k| || ≥ || |i - k| || = d(**s**[i], **s**[k]).
Suppose X is the [[https://en.wikipedia.org/wiki/Indicator_function|indicator function]] (characteristic function) for the set S(|**s**[a] - **s**[b]|, |a - b|), Y for the set S(|**s**[b] - **s**[c]|, |b - c|), and Z for the set S(|**s**[a] - **s**[c]|, |a - c|), which we may regard as vectors in ℝ^**P**. Let J be the **P**-dimensional vector [1, 1, ..., 1] of all 1s. Then what we wish to prove may be rewritten **P** - Z.J  ≤ (**P** - X.J) + (**P** - Y.J). This may be rewritten again as Z.J  ≥ (X + Y - J).J. Every index contributing to X.Y counts as one of Z, and hence Z.J ≥ X.Y. The vector X + Y - J is 1 at an index where both X and Y are 1, is -1 when neither is 1, and 0 otherwise. Hence (X + Y - J).J is X.Y - (J - X).(J - Y), and so is less than or equal to X.Y, and hence less than or equal to Z.J.


These properties mean that the structure metric defines a //finite metric space//. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.
These properties mean that the structure metric defines a //finite metric space//. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.
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&lt;br /&gt;
&lt;br /&gt;
5. d(a, c) ≤ d(a, b) + d(b, c)&lt;br /&gt;
5. d(a, c) ≤ d(a, b) + d(b, c)&lt;br /&gt;
If an interval in the interval class of a equals &lt;strong&gt;s&lt;/strong&gt;[a] and an interval in the interval class of b equals &lt;strong&gt;s&lt;/strong&gt;[b], then their product, reduced modulo the interval of equivalence &lt;strong&gt;O&lt;/strong&gt; which is  &lt;strong&gt;s&lt;/strong&gt;[&lt;strong&gt;P&lt;/strong&gt;], will be &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt;. Hence to get an interval in the class of &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; other than &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; as a product, either the interval in the class of &lt;strong&gt;s&lt;/strong&gt;[i] must be other than &lt;strong&gt;s&lt;/strong&gt;[i], or the interval in the class of &lt;strong&gt;s&lt;/strong&gt;[j] must be other than &lt;strong&gt;s&lt;/strong&gt;[j]. If always only one of the intervals is different than the defining interval for its class, then || &lt;strong&gt;s&lt;/strong&gt;[i + j] mod &lt;strong&gt;O&lt;/strong&gt; || equals ||&lt;strong&gt;s&lt;/strong&gt;[i]|| + ||&lt;strong&gt;s&lt;/strong&gt;[j]||. However, there may be overlap, so that the first interval is not in the class for &lt;strong&gt;s&lt;/strong&gt;[i] and the second not in the class for &lt;strong&gt;s&lt;/strong&gt;[j], so that the count is double on the right hand side. In any case, we get the inequality. Now d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[j]) + d(&lt;strong&gt;s&lt;/strong&gt;[j], &lt;strong&gt;s&lt;/strong&gt;[k]) = || |i - j| || + || |j - k| || ≥ || |i - k| || = d(&lt;strong&gt;s&lt;/strong&gt;[i], &lt;strong&gt;s&lt;/strong&gt;[k]).&lt;br /&gt;
Suppose X is the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Indicator_function" rel="nofollow"&gt;indicator function&lt;/a&gt; (characteristic function) for the set S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[b]|, |a - b|), Y for the set S(|&lt;strong&gt;s&lt;/strong&gt;[b] - &lt;strong&gt;s&lt;/strong&gt;[c]|, |b - c|), and Z for the set S(|&lt;strong&gt;s&lt;/strong&gt;[a] - &lt;strong&gt;s&lt;/strong&gt;[c]|, |a - c|), which we may regard as vectors in ℝ^&lt;strong&gt;P&lt;/strong&gt;. Let J be the &lt;strong&gt;P&lt;/strong&gt;-dimensional vector [1, 1, ..., 1] of all 1s. Then what we wish to prove may be rewritten &lt;strong&gt;P&lt;/strong&gt; - Z.J  ≤ (&lt;strong&gt;P&lt;/strong&gt; - X.J) + (&lt;strong&gt;P&lt;/strong&gt; - Y.J). This may be rewritten again as Z.J  ≥ (X + Y - J).J. Every index contributing to X.Y counts as one of Z, and hence Z.J ≥ X.Y. The vector X + Y - J is 1 at an index where both X and Y are 1, is -1 when neither is 1, and 0 otherwise. Hence (X + Y - J).J is X.Y - (J - X).(J - Y), and so is less than or equal to X.Y, and hence less than or equal to Z.J.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
These properties mean that the structure metric defines a &lt;em&gt;finite metric space&lt;/em&gt;. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.&lt;br /&gt;
These properties mean that the structure metric defines a &lt;em&gt;finite metric space&lt;/em&gt;. This is a structure which has gained a certain amount of attention, particularly in terms of applications in fields requiring data analysis with an eye to similarities and differences.&lt;br /&gt;
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