Cardinality of a set within a set

Question

Given a set $B = \{ \{1,4\}, a, b, \{\{3, 4\}\}, \{\emptyset\} \}$, find the cardinality of $B$

my answer is $5$, and it is from $\{1,4\}, a, b, \{\{3, 4\}\}, \{\emptyset\}$, which is $5$ elements However I'm unsure if that is the correct answer because $\{\{3, 4\}\}$ has a set $\{3,4\}$ within a set.

The resource I found on-line says that: Given $F = \{ \emptyset, \{\emptyset\}, \{\{\emptyset\}\} \}$, the cardinality of $F = 3$, where $\emptyset$ is one, $\{\emptyset\}$ is one, and $\{\{\emptyset\}\}$ is another one.

Therefore I figure $\{\{3,4\}\}$ should also be counted as one cardinal as well.

Please give me some advice, thanks in advance.

Answer 1

You are correct.

The only way you could be wrong (in which case this would be a trick question) is if $a$ or $b$ was equal to one of the other elements of the set, e.g. $a = b$ or $a = \{\{3,4\}\}$.

Answer 2

No matter what the elements of the set are, they only count once when determining cardinality. You determined the cardinality of your set correctly using this guiding principle.

Answer 3

You are correct.
To be more specific, the cardinality of $B$ is $5$, the cardinality of $\{\{3,4\}\}$ is $1$ and the cardinality of $\{3,4\}$ is $2$.