\begin{align} . \end{align} \begin{align} \text{Why is } \emptyset \neq\{\emptyset\}? \end{align}
\begin{align} . \end{align}
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2$\begingroup$ The lhs is an empty set of “points” whereas the RHS is a non-empty set of sets $\endgroup$ – Mathemagical Nov 9 '17 at 8:08
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$\begingroup$ What does a non-empty set mean? $\endgroup$ – Tsangares Nov 9 '17 at 8:14
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3$\begingroup$ It helps if you view sets as packages: For instance {1} is a package that contains the number one, and {2,{1}} is a box that holds 2 AND another box that contains 1 (so a box in a box). Here {1} isn't the same as {{1}} because you have to unpack once in the first case and twice in the second, equivalent to being an element of... or a subset of... . Like that $\emptyset$ means a box that is empty (and holds no elements) and {$\emptyset$} means a box that holds an empty box. In this case the outer box contains one element (an empty box). $\endgroup$ – Michelle_B Nov 9 '17 at 8:16
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Two sets are equal only if they have the same elements.
Part of your confusion is about what elements a set is considered to have. In mathematics containment means direct containment while in every-day speak you allow for an indirect meaning. In everyday language you look into containers inside too.
Take for example your kitchen drawer contains a box of chocolates. In everyday language you'd probably say that there's chocolates in the drawer, but in math language one says there's only a box in the drawer (that box in turn happens to contain chocolates).
When you've then eaten all chocolates, but leaves the box you may consider the drawer to be empty since there's no longer any chocolates. But in math language it's still a box in the drawer (and that box happens to be empty).
Taking this analogy you consider if a drawer with an empty box has the same contents as a drawer with not even an empty box. In math language this is not the case since the first drawer contains a box while the other doesn't (that the box is empty doesn't change this fact).
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The first set ($\emptyset$) has zero elements, the latter ($\{\emptyset\}$) has one element. Therefore they cannot be the same.
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$\begingroup$ Why does the $\{\emptyset\}$ contain one element? $\endgroup$ – Tsangares Nov 9 '17 at 8:17
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$\begingroup$ @Eff I guess his/her question is that why $\{\{\}\}$ doesn't collapse to $\{\}$, because after all $\{\}$ is "nothing". Something like this reasoning. $\endgroup$ – LoMaPh Nov 9 '17 at 8:18
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$\begingroup$ @Tsangares Because $\emptyset\in\{\emptyset\}$. $\endgroup$ – José Carlos Santos Nov 9 '17 at 8:28
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$\begingroup$ @Tsangares Think of the set $\{a\}$. Clearly, this set has one element, namely $a$. Now set $a = \emptyset$ and the same logic holds. $\endgroup$ – Eff Nov 9 '17 at 8:41
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4$\begingroup$ A box that contains an empty box is not the same as an empty box. $\endgroup$ – Gerhard S. Nov 9 '17 at 8:45
