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Difference between revisions of "Delta-sigma-operation"

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A set-theoretic operation, the result of its application to a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d0300103.png" /> of sets can be noted thus:
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A set-theoretic operation, the result of its application to a sequence $(E_n)$ of sets can be noted thus:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d0300104.png" /></td> </tr></table>
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$$\Phi(E_n)=\bigcup_{z\in N}\bigcap_{n\in z}E_n,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d0300105.png" /> is a system of sets of positive integers, called the base of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d0300108.png" />-<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d0300109.png" />-operation. See [[Descriptive set theory|Descriptive set theory]].
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where $N$ is a system of sets of positive integers, called the base of the $\delta$-$\sigma$-operation. See [[Descriptive set theory|Descriptive set theory]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "On operations over sets"  ''Mat. Sb.'' , '''35''' :  3–4  (1928)  pp. 415–422  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Hayes,  R. Probstein,  "Hypersonic flow theory" , New York-London  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "Theory of functions of a real variable and the theory of topological spaces" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.S. Ochan,  "The theory of operations over sets"  ''Uspekhi Mat. Nauk'' , '''10''' :  3  (1955)  pp. 71–128  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Kolmogorov,  "P.S. Alexandroff and the theory of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d030/d030010/d03001010.png" /> operation"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 247–250  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 275–278</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  "On operations over sets"  ''Mat. Sb.'' , '''35''' :  3–4  (1928)  pp. 415–422  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W. Hayes,  R. Probstein,  "Hypersonic flow theory" , New York-London  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  P.S. Aleksandrov,  "Theory of functions of a real variable and the theory of topological spaces" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  Yu.S. Ochan,  "The theory of operations over sets"  ''Uspekhi Mat. Nauk'' , '''10''' :  3  (1955)  pp. 71–128  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  A.N. Kolmogorov,  "P.S. Alexandroff and the theory of the $\delta s$ operation"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 247–250  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 275–278</TD></TR></table>

Revision as of 07:45, 12 August 2014

A set-theoretic operation, the result of its application to a sequence $(E_n)$ of sets can be noted thus:

$$\Phi(E_n)=\bigcup_{z\in N}\bigcap_{n\in z}E_n,$$

where $N$ is a system of sets of positive integers, called the base of the $\delta$-$\sigma$-operation. See Descriptive set theory.

References

[1] A.N. Kolmogorov, "On operations over sets" Mat. Sb. , 35 : 3–4 (1928) pp. 415–422 (In Russian)
[2] W. Hayes, R. Probstein, "Hypersonic flow theory" , New York-London (1959)
[3] P.S. Aleksandrov, "Theory of functions of a real variable and the theory of topological spaces" , Moscow (1978) (In Russian)
[4] Yu.S. Ochan, "The theory of operations over sets" Uspekhi Mat. Nauk , 10 : 3 (1955) pp. 71–128 (In Russian)
[5] A.N. Kolmogorov, "P.S. Alexandroff and the theory of the $\delta s$ operation" Russian Math. Surveys , 21 : 4 (1966) pp. 247–250 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 275–278
How to Cite This Entry:
Delta-sigma-operation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Delta-sigma-operation&oldid=32854
This article was adapted from an original article by A.G. El'kin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article