These statements look correct to me.AxiomOfChoice said:Let b be a real number. Correct me if I'm wrong, but it seems that:
(1) The interval (b,b) is empty, as are the intervals (b,b] and [b,b).
(2) The interval [b,b] consists of a single point (namely, b).
The main difference between these two intervals is that (b,b) is an open interval, meaning that the endpoints b are not included in the interval, while [b,b] is a closed interval, meaning that the endpoints b are included in the interval. In other words, (b,b) represents all real numbers between b and b, but not including b, while [b,b] represents all real numbers between b and b, including b.
An example of (2,5) would be all real numbers between 2 and 5, but not including 2 and 5. An example of [2,5] would be all real numbers between 2 and 5, including 2 and 5.
(b,b) and [b,b] intervals are commonly used in calculus and real analysis to define and represent limits, continuity, and differentiability of functions. They are also important in understanding and solving problems involving inequalities and intervals on the real number line.
No, there are other types of intervals such as half-open intervals (a,b] or [a,b), which include one endpoint but not the other, and unbounded intervals (a,∞) or (-∞,b), which extend infinitely in one direction.
On a number line, (b,b) would be represented by an open circle at b and b, with a dashed line connecting them, while [b,b] would be represented by a closed circle at b and b, with a solid line connecting them. In a coordinate plane, (b,b) would be represented by a dotted vertical line at b, while [b,b] would be represented by a solid vertical line at b.