Does the cardinality of a Null set is same as the cardinality of a set containing single element? If a set A contains Null set as its subset, then the null set is taken into account to calculate the cardinality of set A or not?
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$\begingroup$The cardinality of the emptyset is $0$ whereas the cardinality of a finite set is the number of elements in the set.
More precisely, there's a bijection between a finite set and a set with the form $\left\{0,... ,k-1\right\}$ for $k\in\mathbb{N}$. Then we say the cardianlity of this set is $k$.
Of course, a set may contain other sets (including the empty set). Every set contained is a member as much as a number would be, so in your case: $\left| \left\{ \emptyset \right\} \right| =1 $ since the set contains only one member which is the empty set.
$\endgroup$ 0 $\begingroup$Simply said: the cardinality of a set S is the number of the element(s) in S.
Since the Empty set contains no element, his cardinality (number of element(s)) is 0.
If a set S' have the empty set as a subset, this subset is counted as an element of S', therefore S' have a cardinality of 1.
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