another discrete math question. I was watching this video about Cartesian products, power sets and cardinality when a thought occurred to me. If the set S were arbitrary, would its Cartesian product be a subset of the power set? In other words would this statement be true:
$S \times S \subseteq \mathcal{P}(S)$
I've looked around on the site to find a similar question, but any questions I found involved the Cartesian product of two or more sets, not one. I think the statement is false because with an arbitrary set, we don't know what the elements are, so there's a chance the statement is false by default. Any thoughts?
Best Answer
It is not true for most sets $A$, because the elements of $A\times A$ are ordered pairs, and an ordered pair of elements from $A$ is not a usually set of elements of $A$ (which it would have to be in order to be a member of $\mathcal P(A)$.
In axiomatic set theory, where sets are the only things that exist, an ordered pair must be represented by a particular set -- the most common convention is to consider $(a,b)$ to "really" be an abbreviation for $\{\{a\},\{a,b\}\}$.
When this convention is used, there are a few particular cases where $A\times A$ is indeed a subset of $\mathcal P(A)$.
One example of this is $A=\varnothing$, in which case $A\times A=\varnothing$ which is a subset of anything. A different example is $A=H_\omega$, the set of hereditarily finite sets (which means basically the sets that can be written in finite space with only the symbols
{,}, and,, such as "$\{\{\},\{\{\}\},\{\{\},\{\{\}\}\}\}$").But in both of those cases $A\times A\subseteq \mathcal P(A)$ is more of an accident than something that tells us something interesting and useful about the sets in question.