Is the set {a} equivalent to the set {{a}}? Assuming the sets are immutable, is the set {a} equivalent to the set {{a}}?
It's a simple question but for some reason I haven't been able to find a definite answer online. Thanks very much
 A: Two sets are said to be equivalent, if they have the same number of elements. In this question $\{a\}$ has 1 element and $\{\{a\}\}$ also has 1 element. So, without any doubt the two are equivalent.
A: They might be equivalent depending on what equivalence means in this context, but they certainly aren't equal, which we can see by looking at the axiom of extensionality,
$$\forall A \forall B (\forall x (x ∈ A \leftrightarrow x ∈ B) \leftrightarrow A = B)$$
In other words, if $A = \{ a \}$ and $B = \{ \{ a \} \}$, then $a ∈ A \leftrightarrow a ∈ B$ is false because $ a ∉ B$. Similarly, we could say $\{ a \} ∈ A \leftrightarrow \{ a \} ∈ B$ is false because $\{ a \} ∉ A$, but in either case $A = B$ is false because $\forall x (x ∈ A \leftrightarrow x ∈ B)$ is false. 
A: The notation $\{a\}$ is commonly taken to stand for a set $X$ with the following property:
$$ \forall x\colon x\in X\leftrightarrow x=a.$$
Accordingly, the notation $\{\{a\}\}$ is commonly taken to stand for a set $Y$ with the following property:
$$ \forall x\colon x\in Y\leftrightarrow x=X,$$
where $X$ is as above.
In the perhaps most commonly used set theory, ZF (with or without choice), one axiom is the Axiom of Foundation
$$ \forall a\colon (\exists b\colon b\in a)\to\exists b\colon (b\in a\land\forall c\colon (c\in b\to c\notin a))$$
or "every non-empty set is disjoint to one of its elements".
Applying the Axiom of Foundation to $Y$ above, we conclude that the only element  $X$ must be disjoint from $Y$, which again implies that the only element $a$ of $X$ is not an element of $Y$. This, together with the Axiom of Extensionality
$$\forall x\colon\forall y:(x=y\leftrightarrow\forall z\colon (z\in x\leftrightarrow z\in y) $$
lets us conclude $X\ne Y$, i.e.,
$$\{a\}\ne\{\{a\}\}. $$
A: $ \{a\} $ is a set which contains one element $ a$ which could be also a set.
$\{\{a\}\} $ is a set whose unique element is $ \{a\} $.
Depending on the context, an object of the set theory, can be seen as an element or a set.
A: Not in general.  Consider Zermelo's ordinals, $\{\},\{\{\}\},\{\{\{\}\}\},\dots$.  This is how he defined the naturals.
