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I'm interested in the problem from this post, only I would like to return a tuple instead of a set.

Specifically, if $\theta = (3, 4, 2)$ and $E = \{1, 2, 3\}$, then I'd like to get $(3, 2)$ where the order of the intersecting elements is preserved from the original tuple.

Based on the linked post above, I feel like

$(3, 2) = (\theta_i|\theta_i \in \theta, \theta_i \in E)$

wouldn't work. Is there a nice notation that would allow me to achieve what I'm looking for? I also have seen this answer, but I'm afraid of putting too much reliance on my audience knowing the details of set notation.

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I think that what you suggest would work, but recommend not relying on anything like it. Don't make the reader work harder than necessary. Invent your own notation, perhaps $$ \theta \sqcap E $$ along with a definition (mostly in words) and an example like the one you provided here.

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