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If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the various components; see here.

Is there an analogue of the term 'summand' for unions and intersections?

That is, for $\bigcup\limits_{i=1}^n A_i$ and $\bigcap\limits_{i=1}^nA_i$, is there a term which refers to the sets $A_i$?

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I don't recall ever hearing a particular term for these. –  Asaf Karagila Sep 5 '13 at 7:16
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I always wondered what these words were for products. Good link. –  pbs Sep 5 '13 at 7:25
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@pbs For products, I'd call them factors. –  alex.jordan Jun 10 at 4:52
    
I don't know if I'd use \displaystyle in this post; I'd have opted for \limits instead. But there's no need to edit. It's just a reminder of that option, for future posts. –  Asaf Karagila Jun 10 at 7:24
    
@AsafKaragila: Thanks, I had only used \limits for using $\lim$ inline. I know I didn't have to edit but it does look much better. –  Michael Albanese Jun 10 at 8:45

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