1. Let a and b be fixed real numbers such that a < b on a number line. The different types of intervals we have are

1. The open interval (a, b): We define an open interval (a, b) with end points a and b as a set of all real numbers "x", such that a < x < b. That is, the real number x will be taking all the values between a and b. An important point to consider in this case is the type of brackets used. Generally open intervals are denoted by ordinary brackets ( ).

2. The closed interval [a, b]: We define a closed interval [a, b] with end points a and b as a set of all real numbers "x", such that a  ≤  x   ≤   b. In this case the real number x will be taking all the values between a and b inclusive of the end points a and b. Generally closed intervals are denoted by [  ] brackets.

3. The half open interval [a, b): We define a half open interval [a, b) with end points a and b as a set of all real numbers "x", such that a   ≤   x < b. In this case the real number x will be taking all the values between a and b, inclusive of only a but not b.

4. The half open interval (a, b]: We define a half open interval (a, b] with end points a and b as a set of all real numbers "x", such that a < x   ≤   b. In this case the real number x will be taking all the values between a and b, inclusive of only b but not a.