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If I wanted to write the repeated union of sets $A_1 \cup ... \cup A_n$ I could write this succinctly using big union notation as $$ \bigcup_{i=1}^n A_i $$

How can I best express that all $A_i = ... = A_n$ are equal to each other using 'big equality' notation?

The best I could come up with was using either a big 'for all' $$ {\Large \forall}_{i,j}^{n}\; A_i = A_j $$

or a big equal sign $$ {\Large =}_{i}^n\; A_i $$

The first option is not as succinct, and the second option may not be clear to a reader. Is there a better or more recognised notation for repeated equality?

Jedf
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    For what it's worth, I've often thought it would be convenient to have such notation, especially with a "cyclic" variant so that one could write, say, $$\underset{cyc}{\huge{=}}; a^2+3b c \qquad\text{for}\qquad a^2 + 3b c ;=; b^2 + 3c a ;=; c^2 + 3a b$$ – Blue Oct 21 '22 at 08:44

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Most people will understand "all $A_i$ are equal". There is no need to invent a new notation for this.

"Big operator" notation was originally designed for, and thus works best with, binary operators taking two objects of type $A$ and returning an object of type $A$. Equality is a predicate, taking two objects of type $A$ and returning a boolean value (which is usually not of type $A$). Hence a big operator notation is awkward for equality.

You could write for "all $A_i$ are equal" $$\bigcup_iA_i=\bigcap_iA_i$$ but this is less clear than the worded form.

Parcly Taxel
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  • I was originally wanting to include this statement inside a larger expression using only mathematical notation, but I suppose short phrases are fine if they are clearer. – Jedf Oct 24 '22 at 02:25