Cardinality of $A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$

Question

Given three sets $A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$ what are the cardinalities of those sets ? Obviously cardinality of $A$ is $0$ and cardinality of $B$ is $1$, but I am not sure about set $C$, because some sources say that cardinality of such set is $2$. Can you please clarify this to me ?

Answer 1

$$A = \varnothing, B = \{ \varnothing \}, C = \{\{\varnothing\}\}$$

The way you have it $B$ and $C$ both have cardinality of $1$

My guess is that you wanted $$C = \{ \varnothing ,\{\varnothing\}\}$$

Which has cardinality $2$.

Answer 2

The cardinality of $A$ is 0. The cardinality of $B$ and $C$ is both 1. For $B$ it is clear. $C$ just contains one element as well. This is the set that contains the emptyset $\emptyset$. Namely $B$. One could write $C=\{B\}$, which makes it clear.

Answer 3

The power set of $\emptyset$ is indeed $\lbrace \emptyset \rbrace$, as the only subset of $\emptyset$ is $\emptyset$ itself.

The power set of $\lbrace \emptyset \rbrace$ indeed does have two elements: $\emptyset$ and the set itself: $\lbrace \emptyset \rbrace$, thus making it $\lbrace \lbrace \emptyset \rbrace, \emptyset \rbrace$.