Set notation
From Why start at x, y, z
When defining a set, something is normally put between curly braces. There are a few different conventions for what can go inside:
- All the elements of the set, separated by commas: \( \{a,b,c, \ldots\} \)
- Set-builder notation: a name for a general element of the set, then a condition that all elements must satisfy, with either a vertical bar or a colon in between: \( \{ x \mid P(x) \} \) or \( \{ x : P(x) \} \).
Piper H asks[1], in the notation \( \{ a_1, a_2, \ldots \} \), where the elements inside are a sequence, are you making an implicit claim that the elements are countable?
Colons are used in set notation to indicate a condition the elements must satisfy. Colons are also used when defining functions to show which sets a function maps between (eg \(f:\mathbb{R}\to\mathbb{R}\). This can cause ambiguities when defining a set containing functions, for example:
$$\{f:\mathbb{R}\to\mathbb{R}:f>0\}$$
