1. ## empty set question

what is the cardinality of this sets:

$\displaystyle a.) \left\{{\varnothing}, \left\{{\varnothing}\right\}, \left\{{\varnothing}, \left\{{\varnothing}\right\}\right\}\right\}$

2. Originally Posted by TheRekz
what is the cardinality of this sets:

$\displaystyle a.) ({\varnothing, ({\varnothing}), ({\varnothing, ({\varnothing})}}))$
The elements are:
$\displaystyle \varnothing , \{ \varnothing \} , \{\varnothing,\{\varnothing \} \}$

1+1+1=3

3. Originally Posted by TheRekz
what is the cardinality of this sets:
$\displaystyle a.) ({\varnothing, ({\varnothing}), ({\varnothing, ({\varnothing})}}))$
First, I must say that I do not think that you are asking the question that you mean to ask.
Because, I see only one element of the set of triples, 3-tuples.
Now, I am of course aware that a set of ordered pairs is defined as sets of sets: the pair $\displaystyle \left( {a,b} \right) = \left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}$
And we define the triple $\displaystyle \left( {a,b,c} \right) = \left( {\left( {a,b} \right),c} \right)$.
Now, I am of course aware that a set of ordered pairs is defined as sets of sets: the pair $\displaystyle \left( {a,b} \right) = \left\{ {\left\{ a \right\},\left\{ {a,b} \right\}} \right\}$
And we define the triple $\displaystyle \left( {a,b,c} \right) = \left( {\left( {a,b} \right),c} \right)$.