Cardinality of a set $\{A,B\}$ $A$ is a subset of $B$, which is a subset of $S$

Question

Let's say that $A$ is a subset of $B$ and be is a subset of a set $S$ of $n$ elements.

How big is the set $\{(A,B)\}$ then.

Answer 1

Assuming $A \neq B$, then there are exactly two elements in $\{A, B\}$, elements which happen to be sets $A, B$. Hence, the cardinality of $\{A,\,B\}$ is $2$. I.e., $|\{A, \,B\}| =2.$ Else, we can only say $|\{A, \,B\}| \leq 2$.

The set $\{(A, B)\}$ has one element, (cardinality 1), which happens to be an ordered pair of sets.

The information about the relation between sets $A, B,$ and between $B, S$ and the elements of $S$ is irrelevant: that information is what they call a "red herring."

Answer 2

The set $\{A,B\}$ has at most two elements, and if $A\neq B$ then it has exactly two.

The set $\{(A,B)\}$ has exactly one element, the ordered pair $(A,B)$.

Answer 3

Perhaps you are asking to find $|N|$ where $N = \{(A, B) \mid A \subseteq B \subseteq S, |S| = n\}$ for some integer $n$.

To solve this, let's first fix some $B_0$ and let $m = |B_0|$. Then the number of subsets of $B$ is just $2^m$. Clearly, $0 \leq m \leq n$. So $|N| = \sum_{i=0}^{n}{2^i}=2^{n+1}-1$

Answer 4

It depends on whether you mean the set in the title of your question, or the set in the question itself.

As $A \subseteq B \subseteq S$, $A$ and $B$ are elements of $\mathcal{P}(S)$, the power set of $S$. So $\{A, B\} \subseteq \mathcal{P}(S)$. This set has cardinality two, unless (as Chris Eagle pointed out in a comment on another answer) $A = B$, in which case $\{A, B\} = \{A\}$ which has cardinality one.

The notation $(\cdot, \cdot)$ is used to denote an ordered pair, which is an element of the cartesian product of two sets. Again, $A, B \in \mathcal{P}(S)$, so $(A, B) \in \mathcal{P}(S)\times\mathcal{P}(S)$, hence $\{(A, B)\} \subseteq \mathcal{P}(S)\times\mathcal{P}(S)$. In particular, $\{(A, B)\} \neq \{A, B\}$. The former set has cardinality one, regardless of whether or not $A = B$.