# How do you simplify $\cup\{A, \cup A\}$?

I want to simplify $$\cup\{A, \cup A\},$$ and also $$\cup\cup\{A, \cup A\},$$ so forth. I thought $$\cup\{A, \cup A\}$$ would not be simplified more. To say this in plain english, this union is a set that contains 1) all elements in $$A$$ and 2) all elements in all elements in $$A,$$ which I am not sure how to simplify. Then $$\cup\cup\{A, \cup A\}$$ is a set that contains all elements in the first union that I just described. Is it possible to simplify this union more? Thanks for your help.

• Sorry I didn't see what I posted was included in your post. Yes, I think it is correct and that It cannot be simplified any further. – JustDroppedIn Jun 10 at 17:15
• No worries! Thank you very much :) Do you think it'd be possible to simplify the second set? I thought not, but I wanted some more insights. – lifeisfun Jun 10 at 17:15
• What kind of notation is this? Is $\cup\{A,\cup A\}$ simply the set $A\cup\bigcup_{x\in A} x$? – Evangelos Bampas Jun 10 at 18:05
• @EvangelosBampas : This notation is used when set theory is not merely used in some other areas of mathematics, but rather is done for the purpose of understanding set theory. $$\underbrace{\quad \cup A \quad}_\text{set theorists' notation} = \underbrace{ \quad \bigcup_{x\in A} x \quad }_\text{common mathematicians' notation}$$ – Michael Hardy Jun 10 at 19:17
• @MichaelHardy: Except for the fact that one should still write $\bigcup A$ and not $\cup A$. – Asaf Karagila Jun 11 at 9:16

You might or might not consider it a simplification, but, of course, $$\bigcup \left\{A, \bigcup A\right\} = A \cup \bigcup A.$$