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I'm ask to decide if there exists $2$ elements $a$,$b$ in a set such that $a+b =8$.

It seems to me that many understand that $a$ and $b$ need to be two different elements. How should I understand "$2$" in this case? The assumption $a\neq b$ is not written, but "$2$ elements" could also indicate that the cardinality of $a$,$b$ has to be $2$.

What to do?

MphLee
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user11775
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    Such formulations are often a source of ambiguity and misinterpretation. It might be clearer to talk about either "distinct elements $a,b$ of $S$ with $a+b=8$" or "(possibly identical) elements $a,b\in S$ with $a+b=8$" instead of "two elements $a,b$ of $S$ with $a+b=8$". But note that "pairwise distinct" would be exaggerated (even in the case of more than two) - distinct objects are alrady "pairwise distinct". In a similar fashion, I often find myself pointing out that writing $X={a,b}$ does notimply that $|X|=2$. – Hagen von Eitzen May 01 '13 at 20:46

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The precise meaning depends on the precise formulation. If one was sufficiently formalist to write the question with quantifiers in the form $\exists a,b \in S :\ a + b = 8$, then $a= b$ would definitely be allowed. Likewise, if one wanted $a$ and $b$ to be distinct, one would have to write this explicitly, and the problem would disappear.

In most practical situations, if the problem is presented in this form, the natural thing would be to ask for notation be made precise, since it is definitely not the case that there is a standard interpretation. I think that depending on context, either of the interpretations could be taken as "obviously right": For instance, a lot of people might express pigeonhole principle by saying that if $n+1$ objects are partitioned into $n$ holes then there are objects $a,b$ that land in the same hole. Noone would ask if they can be the same - of course they can't. Cases when equality is allowed are, I think, more common, but I can't think of a convincing example right now.

The following rule of thumb appears to be valid in the first approximation: if letting the two elements be equal makes the statement trivial, then almost surely they are supposed to be distinct. Likewise, if requiring $a,b$ to be distinct makes the claim (obviously) untrue, then almost surely they do not have to be distinct.

It seems that the word "two" only adds to the confusion. It could be either an attempt to say that $a,b$ are distinct, or a grammatical hint to make sentence easier to parse. It does not look conclusive to me.

Jakub Konieczny
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