Question

For each of these intervals, list all its elements or explain why it is empty. a) [a,a]

b) [a,a]
c) (a,a]
d) (a,a)
e) (a,b), where a>b
f) [a,b], where a>b

Answer 1

In interval notation, each interval is analyzed step-by-step, listing the elements or explaining why it is empty. Both inclusive and exclusive intervals are covered, and specific examples are given to illustrate the concepts.

Step-by-step explanation:

a) [a,a]: This interval contains only one element, which is a. So, the interval is not empty and its element is a.

b) [a,a]: Similar to the previous interval, this interval also contains only one element, which is a. So, the interval is not empty and its element is a.

c) (a,a]: This interval is empty because there are no elements between a and a. The left endpoint a is included, but the right endpoint a is not.

d) (a,a): This interval is also empty because there are no elements between a and a. Neither endpoints a are included in this interval.

e) (a,b), where a>b: This interval is not empty because it contains all the elements between a and b (excluding a and b). For example, if a=5 and b=3, the interval would be (5,3) and its elements would be 4, 3.9, 3.8, and so on.

f) [a,b], where a>b: This interval is empty because a is greater than b, meaning there are no elements between a and b. The left endpoint a is included, but the right endpoint b is not.