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this post is a sub-topic of that post, which seems to be discussing without a common accepted foundation.

although this post has not yet got much discussion, I prefer to use this () notation to denote the sequence.

given the sequence of the prime numbers $(2, 3, 5, 7, 11, 13, 17, ...)$

and

the sequence of the even numbers $(..., -6, -4, -2, 0, 2, 4, 6, ...)$

is the intersection of these 2 sequences a number 2 or a sequence of a single number (2) or a set (I'll pass the last one, anyway, I have no the right answer, so I list all of the possibilities)?

Asaf Karagila
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JJJohn
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  • It's the single number $2$. Or the set consisting of the single number $2$, not sure what distinction you hope to draw there. – lulu Aug 01 '19 at 16:34
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    The intersection of the sets of prime numbers and of even numbers is the set ${2}$. The intersection of two sequences is not defined - since these are'n sets. – Hans Engler Aug 01 '19 at 16:40
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    If you want to be pedantic., a sequence is technically a function, and a function is technically a set... so these are sets in a sense... The first could be the set ${(0,2),(1,3),(2,5),(3,7),(4,11),\dots}$ and the second could be the set ${\dots,(-3,-6),(-2,-4),(-1,-2),(0,0),(1,2),\dots}$. Under this interpretation, the intersection is empty. The intersection under this interpretation would be the set of tuples of position and value such that in both sequences the same value occurs at the same position in each. – JMoravitz Aug 01 '19 at 16:50

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Strictly speaking, in the context of your question, those sequences are just sets, since the order doesn't matter. The intersection of two sets is always a set, even if it happens to be a set with just one element. So you should say "$2$ is the only number in the intersection".

In practice you won't cause confusion if you say "the intersection is $2$". Better just to say "$2$ is the only even prime" with less formality.

If you really did want to talk about the intersection of sequences, what might you mean by the intersection of $(1,2,3)$ and $(4,5,1)$? Each contains $1$, but at a different place.

Ethan Bolker
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  • Also a set with a single element can be called a singleton, that can help with distinguishing the number $2$ and the singleton ${2}$. – Cryme Aug 02 '19 at 10:20