# Set notation

• The symbol "" ("\in" in LaTeX) means "is in" or just "in". For example, , but .
• Curly braces usually indicate a set. I have already used to denote the set containing 1, 2, and 3.
• There is a "set comprehension" syntax that is useful for building other sets. For example, if , we can write . This should be read as "the set of all elements in the set such that ." This is another way of writing . You can use any description (as long as it is clear an unambiguous) in the predicate (the part after the vertical bar). For example .
• Informally, one can describe a set, for example: is an informal way to write . Be careful with this notation, for two reasons:
1. It can be hard to unambiguously determine whether something is in the set or not. Is "tangerine" in ?
2. it is unclear whether is the set containing all colors, or the set containing the set of all colors.
• A much better approach is to describe the set in English: Let be the set of all colors. In general, the test of whether you've given a good definition is not the notation you use, but whether the definition is clear and unambiguous. Sometimes notation helps with this, but other times it makes your writing less clear.
• The symbol (LaTeX \emptyset) stands for the empty set (which contains no elements).