Question

What is an open interval?

Answer 1

See explanation...

Explanation:

If `a` and `b` are Real numbers with `a < b` then `(a, b)` is used to denote the numbers which lie strictly between `a` and `b`. This is called "the open interval `a`, `b`".

In symbols we could write: `(a, b) = { x in RR : a < x < b }`

This reads `(a, b)` is the set of elements `x` in the set of Real numbers (`RR` for short) such that `a < x` and `x < b`.

When we want to talk about all the Real numbers we may write:

`(-oo, +oo)`

The symbols `-oo` (minus infinity) and `+oo` (plus infinity) are not really numbers. You can picture them as being at the extreme left and right ends of the Real number line. Any Real number `x` satisfies:

`-oo < x < +oo`

If we want to talk about any number greater than `5`, we can write:

`x in (5, +oo) " "` (`x` is in the open interval "`5` to plus infinity")

If we want to talk about any number less than `5`, we can write:

`x in (-oo, 5) " "` (`x` is in the open interval "minus infinity to `5`")

If `z != 5` then either `z < 5` so `z in (-oo, 5)` or `z > 5` so `x in (5, oo)`.

The amalgamation of these two sets is called the union and is denoted:

`(-oo, 5) uu (5, +oo)`