What are 'intervals'? Explain with examples (i.e sets).

The intervals are sets of numbers. All the numbers
that have the property to be located between the endpoints of the interval are elements
of the set.

Examples:

The
open interval (-1;5) is the set of real numbers x that have the
property:

-1 < x <
5

All the real numbers located between -1 and 5, excluding
-1 and 5, are the elements of the interval (-1,5).

(-1,5) =
{x belongs to R / -1 < x < 5 }

The closed
interval [-5,10] is the set of real numbers x that have the
property:

-5 =< x =<
10

[-5,10] = {x belongs to R / -5 =< x =<
10}

All the real numbers located between -5 and 10,
including -5 and 10, are the elements of the interval
[-5,10].

There are included also the intervals that have an
open endpoint and a closed endpoint:

(a,b] or
[a,b)

That means  the
following:

(a,b] = {x belongs to R / a< x =<
b}

[a,b) = {x belongs to R / a =< x <
b}

There is the possibility for one of the endpoints to be
infinite. This means that the set has no limit to the left or to the right, depending
what endpoint is infinite.