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I found in a text, the following notation:

$\lbrace 1, 2, 3, 4 \rbrace\cap 2= \lbrace 2 \rbrace$

To my knowledge, a set and an element cannot be concatenated by a set intersection i.e., the set intersection is defined for two sets, but neither for two elements nor for combining a set and an element. Therefore, the correct notation i think for the above line is

$\lbrace 1, 2, 3, 4 \rbrace\cap \lbrace 2\rbrace= \lbrace 2 \rbrace$.

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    $\begingroup$ You are correct. The first notation is nonsense. $\endgroup$ – Matthew Daly Nov 13 '19 at 13:48
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You are completely right, and this is most likely a typo in the text.

Sometimes, the natural numbers themselves do denote a set. So for example, 2 would denote the set $\{0, 1\}$. In general, $n = \{0, 1, \ldots, n-1\}$. This is under the interpretation of natural numbers as ordinal numbers. In that case, we could make sense of the first intersection and we would have: $$ \{1,2,3,4\} \cap 2 = \{1,2,3,4\} \cap \{0,1\} = \{1\}. $$

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