Information AboutSubset |
| CATEGORIES ABOUT SUBSET | |
| basic concepts in set theory | |
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In Mathematics , especially in Set Theory , a Set ''A'' is a subset of a set ''B'', if ''A'' is "contained" inside ''B''. The relationship of one set being a subset of another is called '''inclusion''' or '''containment'''. DEFINITIONS If ''A'' and ''B'' are sets and every Element of ''A'' is also an element of ''B'', then:
:or equivalently
If ''A'' is a subset of ''B'', but ''A'' is not Equal to ''B'' (i.e. there exists at least one element of B not contained in ''A''), then
:or equivalently
For any set ''S'', the inclusion relation ⊆ is a Partial Order on the set 2''S'' of all subsets of ''S'' (the Power Set of ''S''). SYMBOLS | ||
| For The | "http://wwwinformationdelightinfo/information/entry/power_set" class="copylinks">Power Set 2<sup>''S''</sup> of a set ''S'', the inclusion partial order is (up to an Order Isomorphism ) the Cartesian Product of ''k'' = ''S'' (the Cardinality of ''S'') copies of the partial order on {0,1} for which 0 < 1 This can be illustrated by enumerating ''S'' = {''s''<sub>1</sub>, ''s''<sub>2</sub>, …, ''s''<sub>''k''</sub>} and associating with each subset ''T'' ⊆ ''S'' (which is to say with each element of 2<sup>''S''</sup>) the ''k''-tuple from {0,1}<sup>''k''</sup> of which the ''i''th coordinate is 1 if and only if ''s''<sub>''i''</sub> is a member of ''T'' |
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