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In mathematics, '''Bondy's theorem''' is a bound on the number of elements needed to distinguish the sets in a [[family of sets]] from each other. It belongs to the field of [[combinatorics]], and is named after [[John Adrian Bondy]], who published it in 1972.<ref>{{Citation
| last=Bondy | first=J. A. | authorlink=John Adrian Bondy
| title=Induced subsets
| journal=[[Journal of Combinatorial Theory, Series B]]
| volume=12
| year=1972
| pages=201–202
| doi=10.1016/0095-8956(72)90025-1
| mr=0319773
}}.</ref>


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In other words, if we have a 0-1 [[matrix (mathematics)|matrix]] with ''n'' rows and ''n'' columns such that each row is distinct, we can remove one column such that the rows of the resulting ''n''&nbsp;×&nbsp;(''n''&nbsp;&minus;&nbsp;1) matrix are distinct.<ref>{{citation
 
| first=Stasys | last=Jukna
  <li>[http://www.snesup-paris13.org/spip.php?article16/ http://www.snesup-paris13.org/spip.php?article16/]</li>
  | title=Extremal Combinatorics with Applications in Computer Science
 
| publisher=Springer
  <li>[http://myextremeauto.com/activity/p/19011/ http://myextremeauto.com/activity/p/19011/]</li>
| year=2001
 
| isbn=978-3-540-66313-3
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}}, Section 12.1.</ref><ref>{{citation
 
| first1=Peter | last1=Clote
  <li>[http://badchicsclub.com/activity/p/28194/ http://badchicsclub.com/activity/p/28194/]</li>
| first2=Jeffrey B. | last2=Remmel
 
| title=Feasible Mathematics II
  <li>[http://www.jstlpf18.com/news/html/?174209.html http://www.jstlpf18.com/news/html/?174209.html]</li>
| publisher=Birkhäuser
 
| year=1995
</ul>
| isbn=978-3-7643-3675-2
}}, Section 4.1.</ref>


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==Example==
Consider the 4&nbsp;×&nbsp;4 matrix
:<math>\begin{bmatrix}
    1&1&0&1\\
    0&1&0&1\\
    0&0&1&1\\
    0&1&1&0
\end{bmatrix}</math>
where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix
:<math>\begin{bmatrix}
    1&0&1\\
    1&0&1\\
    0&1&1\\
    1&1&0
\end{bmatrix}</math>
no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3&nbsp;×&nbsp;4 matrix
:<math>\begin{bmatrix}
    1&1&1\\
    0&1&1\\
    0&0&1\\
    0&1&0
\end{bmatrix}</math>
are distinct. Another possibility would have been deleting the fourth column.


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==Learning theory application==
 
From the perspective of [[computational learning theory]], Bondy's theorem can be rephrased as follows:<ref>{{citation
  <li>[http://bbs.yuanmengjihua.com/forum.php?mod=viewthread&tid=89358 http://bbs.yuanmengjihua.com/forum.php?mod=viewthread&tid=89358]</li>
| last1 = Kushilevitz | first1 = Eyal
 
| last2 = Linial | first2 = Nathan
  <li>[http://www.eagleq.com/bbs/forum.php?mod=viewthread&tid=98551 http://www.eagleq.com/bbs/forum.php?mod=viewthread&tid=98551]</li>
| last3 = Rabinovich | first3 = Yuri
 
| last4 = Saks | first4 = Michael
  <li>[http://182.140.249.57/news/html/?64334.html http://182.140.249.57/news/html/?64334.html]</li>
| doi = 10.1016/S0097-3165(96)80015-X
 
| issue = 2
  <li>[http://www.snesup-paris13.org/spip.php?article16/ http://www.snesup-paris13.org/spip.php?article16/]</li>
| journal = Journal of Combinatorial Theory
 
| mr = 1370141
  <li>[http://www.yaocq.com/news/html/?359952.html http://www.yaocq.com/news/html/?359952.html]</li>
| pages = 376–380
 
| series = Series A
</ul>
| title = Witness sets for families of binary vectors
| volume = 73
| year = 1996}}.</ref>


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This implies that every finite concept class ''C'' has its [[teaching dimension]] bounded by |''C''|&nbsp;&minus;&nbsp;1.
 
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[[Category:Computational learning theory]]
 
[[Category:Theorems in combinatorics]]
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Latest revision as of 04:59, 13 December 2013

In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972.[1]

Statement

The theorem is as follows:

Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct.

In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct.[2][3]

Example

Consider the 4 × 4 matrix

[1101010100110110]

where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix

[101101011110]

no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix

[111011001010]

are distinct. Another possibility would have been deleting the fourth column.

Learning theory application

From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows:[4]

Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C.

This implies that every finite concept class C has its teaching dimension bounded by |C| − 1.

Notes

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