Cardinality of $S=\{A,B\}$ where $A=B=\{1,2,3\}$.

Question

Given a set $S = \{A,B\}$ such that $A = \{1,2,3\}$ and $B = \{1,2,3\}$, what is the cardinality of S?

I know this may seem very trivial, and I am inclined to believe that the answer is $2$; my question comes from the fact that $A = B$. Since both $A$ and $B$ are the same, would that not be the same as having a repeated element, therefore making the answer $1$?

Answer 1

$S$ has one element, so has cardinality $1$. (Although we have two labels for $\{1,2,3\}$, there is still only one such set.) Sets don't have duplicates. (Multisets can have duplicates. If $S$ were a multiset, it would have cardinality $2$.) The set $\{x, x, x, x, x, x, x, x\}$ has one element, $x$.

Answer 2

The cardinality of your the set $S$ would be $1$. Since $A$=$B$, the cardinality is $1$.

Also, sets that are subsets of a set are just items if you will, and can be counted as elements. The only reason the cardinality is not $2$ is because the sets are equal

Answer 3

By the vanishing property, since $A=B$, $S=\{A\}=\{B\}$, hence $|S|=1$.