Used to demonstrate that a variable
can take on any one of a range
of values. If we have an equation where x is allowed to be any real number
between a and b, inclusive
, we write x = [a, b]. If x cannot be a or b, but can be anything in between
, we write x = (a, b). If x can be anything between a and b, including a but not b, we write x = [a, b). Determining what (a, b] might represent is left as an exercise to the reader.
If x is allowed to be anything greater than a, and does not have an upper bound, we write x = (a, ∞). Similarly, if x can be anything less than or equal to a, with no lower bound, we write x = (-infinity, a]. And if x can be any real number, then x = (-∞, ∞). We never write anything like [-infinity, a), because no variable can ever equal ∞ or -∞. These are not actually numbers, just ways of denoting that the bound is beyond the finite numbers.
Note: The assumption is made above that a < b in all cases, because the notation makes no sense otherwise. It should also be noted that rather than writing x = (a,b), math people often use the lower-case epsilon, which means "is an element of": for example, x ε (a, b).