What does this mean in English: $\bigcup\mathcal{P}A = A$ ? I assume it has something to do with unions and powersets but I don't understand the meaning. Neither do I know if the sentence is true or false.
$\endgroup$2 Answers
$\begingroup$In set theory, if $B$ is a set, $\bigcup B$ is the union of all elements(*) of $B$. This is a short-cut for $\displaystyle{ \bigcup_{b\in B}b }$.
And you're correct, $\mathcal P$ stands for the power set, so $\displaystyle{\bigcup{\mathcal P}A=\bigcup_{X\in {\mathcal PA}}X}=\bigcup_{X\subset A}X$. Which is clearly equal to $A$.
For example, if $A=\{1,2\}$, then ${\mathcal P}A=\{ \emptyset,\ \{1\},\ \{2\},\ \{1,2\}\ \}$ and $$\displaystyle{\bigcup{\mathcal P}A=\emptyset \cup \{1\}\cup \{2\}\cup \{1,2\} = A }$$
(*) note that in set theory, all objects, including real numbers, vectors, matrices, curves, functions, sequences, etc... are sets, so it makes sense to take the union of the elements of a set. In your specific context, ${\mathcal P}A$ is a set of sets in the intuitive meaning of sets, so you don't need to bother with the abstraction of set theory.
$\endgroup$ 5 $\begingroup$If $X = \{x_1, \dots, x_n\}$ is a set, then $\bigcup X$ means $x_1\cup \dots \cup x_n$. Similarly, if $X = \{x_1, x_2, \dots\}$ is infinite, then $\bigcup X = x_1\cup x_2\cup \dots$. If the items $x_1, x_2, \dots$ are sets, this makes sense; if they're not sets, it doesn't. This isn't an issue in the context of elementary set theory since, there, everything is a set.
In the case in the question, all the elements of $\mathcal{P}A$ are sets: specifically, they're all the subsets of $A$. So, the meaning of the statement "$\bigcup \mathcal{P}A = A$" is "the union of all subsets of $A$ is $A$", which is true.
$\endgroup$
