Namespaces
Variants
Actions

Difference between revisions of "Empty set"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (→‎References: isbn link)
 
(3 intermediate revisions by 2 users not shown)
Line 1: Line 1:
The set which contains no elements. Notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035590/e0355901.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035590/e0355902.png" />. In other words, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035590/e0355903.png" />. Moreover, any assertion that is always false could be used in this definition instead of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035590/e0355904.png" />. The empty set is a subset of any set.
+
{{TEX|done}}{{MSC|03E}}
 +
 
 +
The set which contains no elements. Notation: $\emptyset$, $\{\}$, $\Lambda$. In other words, $\{x:x\neq x\}$. Moreover, any assertion that is always false could be used in this definition instead of $x\neq x$. The statement $x \in \emptyset$ is always false.  The empty set is a subset of any set. 
 +
 
 +
====References====
 +
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> P. R. Halmos, ''Naive Set Theory'', Springer (1960) {{ISBN|0-387-90092-6}}</TD></TR>
 +
</table>

Latest revision as of 08:09, 18 November 2023

2020 Mathematics Subject Classification: Primary: 03E [MSN][ZBL]

The set which contains no elements. Notation: $\emptyset$, $\{\}$, $\Lambda$. In other words, $\{x:x\neq x\}$. Moreover, any assertion that is always false could be used in this definition instead of $x\neq x$. The statement $x \in \emptyset$ is always false. The empty set is a subset of any set.

References

[a1] P. R. Halmos, Naive Set Theory, Springer (1960) ISBN 0-387-90092-6
How to Cite This Entry:
Empty set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Empty_set&oldid=16423
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article