# Intersection of an element and a set containing that element.

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

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

• You are correct. The first notation is nonsense. – Matthew Daly Nov 13 '19 at 13:48

## 1 Answer

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: $$\{1,2,3,4\} \cap 2 = \{1,2,3,4\} \cap \{0,1\} = \{1\}.$$