Proper Subset
المؤلف:
Kamke, E
المصدر:
Theory of Sets. New York: Dover
الجزء والصفحة:
p. 6,
17-1-2022
2696
Proper Subset
A proper subset 
of a set 
, denoted 
, is a subset that is strictly contained in 
and so necessarily excludes at least one member of 
. The empty set is therefore a proper subset of any nonempty set.
For example, consider a set ![<span style=]()
{1,2,3,4,5}" src="https://mathworld.wolfram.com/images/equations/ProperSubset/Inline6.svg" style="height:22px; width:106px" />. Then ![<span style=]()
{1,2,4}" src="https://mathworld.wolfram.com/images/equations/ProperSubset/Inline7.svg" style="height:22px; width:64px" /> and ![<span style=]()
{1}" src="https://mathworld.wolfram.com/images/equations/ProperSubset/Inline8.svg" style="height:22px; width:22px" /> are proper subsets, while ![<span style=]()
{1,2,6}" src="https://mathworld.wolfram.com/images/equations/ProperSubset/Inline9.svg" style="height:22px; width:64px" /> and ![<span style=]()
{1,2,3,4,5}" src="https://mathworld.wolfram.com/images/equations/ProperSubset/Inline10.svg" style="height:22px; width:106px" /> are not.
REFERENCES
Kamke, E. Theory of Sets. New York: Dover, p. 6, 1950.
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