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Is it technically correct to write "Let $A$ and $B$ be two sets"? or we should write "Let $A,B$ be sets"? Actually I am confused whether by mentioning "two" we are ruling out the possibility of $A=B$. Please suggest!

Anupam
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    I do not think the presence of 'two' affects the implied meaning in any way. – MathIsNice1729 Mar 11 '16 at 09:31
  • I think the logical statement is true irrespective of the language used here; you are not discounting the notion that possibly $A=B$ with the definition of two sets. Remeber that the statement $A=B$ is mealry a consequence of the fact that for two sets $A, B$ we have $A \subseteq B$ and $B \subseteq A$. –  Mar 11 '16 at 09:35
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    You can always say: "Let $A$ and $B$ be two sets, not necessarily distinct", to avoid confusion. –  Mar 11 '16 at 09:39
  • You can also say "a couple of sets" – Marco Disce Mar 11 '16 at 09:56

3 Answers3

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As a reader, one should not make assumptions about mathematical statements which have not been stated. Thus I feel that you are not losing anything by saying "Let $A$ and $B$ be two sets".

If you have any extra conditions which you feel are non trivial for the reader, then you should mention them.

Banach Tarski
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As for me, I think saying $A$ and $B$ are two sets does not rule out the possibility of $A=B$. It just saying these two sets are not always equal to each other, but they can sometimes.

congmingniao
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If you wish to define two sets, $A$ and $B$, saying "let $A$ and $B$ be two sets" does not exclude the possibility that they will equal each other. You can continue the statement as you like, such as (if they are equal) "let $A$ and $B$ be two sets that are equal to each other" (or, and I think this is more broad but possibly more correct) "let $A$ and $B$ be two sets, which are elements of each other."

I hope this helped.

Gil Keidar
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