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.
[Math] What does $\bigcup\mathcal{P}A = A$ mean in English
discrete mathematicselementary-set-theory
Best Answer
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.