Question about the intervals (b,b) and [b,b]

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In summary, the main difference between (b,b) and [b,b] intervals is that (b,b) is an open interval, while [b,b] is a closed interval. (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. These intervals are commonly used in calculus and real analysis to define and represent limits, continuity, and differentiability of functions. They can also be represented on a graph as open or closed circles on a number line, and as vertical lines on a coordinate plane. However, they are not the only types of intervals, as there are also half-open and unbounded intervals.
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AxiomOfChoice
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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).
 
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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).
These statements look correct to me.
 

1. What is the difference between (b,b) and [b,b] intervals?

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.

2. Can you give an example of (b,b) and [b,b] intervals?

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.

3. How are (b,b) and [b,b] intervals used in mathematics?

(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.

4. Are (b,b) and [b,b] intervals the only types of intervals?

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.

5. How can (b,b) and [b,b] intervals be represented on a graph?

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.

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