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Suppose I gave you the collection of sets $\textit{L}=\{{S_1,S_2}\}, S_1 \neq S_2$. I would think that $|\textit{L}| = 2$ because the collection of sets $\textit{L}$ has only two distinct elements. However, the empty set $\emptyset$ is inherently in every collection of sets, no? So should $|\textit{L}|=3$ instead? Does the answer change when I explicitly state that the empty set is in $\textit{L}$? That is, $\textit{L}=\{\emptyset, S_1, S_2\}$.

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    $\begingroup$ The empty set is a subset, not necessarily an element, of any set. The cardinality of a (say, finite) set is the number of elements it contains. $\endgroup$ – anomaly Oct 6 '16 at 17:08
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    $\begingroup$ No. The empty set is a subset of every set (and hence of $L$), but it is not an element of every set. And it’s an element of $L$ (and hence counted towards $|L|$) if and only if $S_1=\varnothing$ or $S_2=\varnothing$. $\endgroup$ – Brian M. Scott Oct 6 '16 at 17:09
  • $\begingroup$ @BrianM.Scott So does that mean the empty set is an element when I specified $\textit{L}=\{\emptyset, S_1, S_2\}$? And hence this L would have cardinality 3? Because is equivalent to me saying $\textit{L} = \{S_1, S_2, S_3\}, S_1 \neq S_2 \neq S_3, S_3 = \emptyset$, right? $\endgroup$ – Bryyo Oct 6 '16 at 17:18
  • $\begingroup$ If $L=\{\emptyset,S_1,S_2\}$, you’ve named three different things, yes? So $|L|=3$. $\endgroup$ – Lubin Oct 6 '16 at 17:25
  • $\begingroup$ @Bryyo357: Yes, assuming that $S_\ne S_2$ and that $S_1$ and $S_2$ are non-empty, $|\{\varnothing,S_1,S_2\}|=3$. $\endgroup$ – Brian M. Scott Oct 6 '16 at 17:31
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The empty set is not inherently in every set. You are getting confused. The empty set is inherently a SUBSET of every set. There's a difference between subset and element. The set {$S_1,S_2$} has cardinality 2. The set {$S_1,S_2,\varnothing$} has cardinality 3 because it has 3 elements.

The empty set is a subset of both of these however: for if, x is in $\varnothing$, then x is in the set A, where A is ANY set. The reason is that it is impossible for x to be in the empty set, because the empty set is empty. Since the hypothesis is guaranteed false, the truth of the conclusion doesn't matter and thus any implication with this as a hypothesis is automatically true.

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