Answer
a) 0
b) 1
c) 2
d) 3
Work Step by Step
a) The set $\emptyset$ contains no elements. So the cardinality of $\emptyset$ is 0.
b) The set $\{ $$\emptyset$$\}$contains exactly one element, namely $\emptyset$. Thus the cardinality of $\{ $$\emptyset$$\}$ is 1.
c) The set $\{ $$\emptyset$ , $\{ $$\emptyset$$\}$$\}$ contains two elements, which are $\emptyset$ and the singleton $\{ $$\emptyset$$\}$. So the cardinality of this set is 2.
d) The set $\{$$\emptyset$, $\{ $$\emptyset$$\}$, $\{ $$\emptyset$ , $\{ $$\emptyset$$\}$$\}$$\}$ contains the three elements $\emptyset$, $\{ $$\emptyset$$\}$, and $\{ $$\emptyset$ , $\{ $$\emptyset$$\}$$\}$. So the cardinality of this set is 3.
