REAL NUMBERS, R

8. INTERVALS

Consider the following set of numbers; 

         A1 = {x| 2 < x < 5}

         A2 = {x| 2 < x < 5}

         A3 = {x| 2 < x < 5} 

         A4 = {x2 < x < 5}

Notice, that the four sets contain only the points that lie between 2 and 5 with the possible exceptions of 2 and/or 5. We call these sets intervals, the numbers 2 and 5 being the endpoints of each interval. Moreover, A1 is an open interval  as it does not contain either end point: A2 is a closed interval as it contains bother endpoints; A3 and A4 are open-closed and closed-open respectively

Notice that in each diagram we circle the endpoints 2 and 5 and thicken (or shade) the line segment between the points. If an interval includes an endpoint, then this is denoted by shading the circle about the endpoint.

Since intervals appear very often in mathematics, a shorter notation is frequently used to designated intervals, Specifically, the above intervals are sometimes denoted by;

          A1 = (2, 5)

          A2 = [2, 5]

           A3 = (2, 5)

           A4 = [2, 5)

Notice that a parenthesis is used to designate an open endpoint, i.e. an endpoint that is not in the interval, and a bracket is used to designate a closed endpoint.