91Ó°ÊÓ

The set of real numbers includes all the numbers that can be found on the number line. This includes both rational numbers, like fractions and integers, such as

• Whole numbers (e.g., 0, 1, 2)
• Fractions (e.g., 1/2, 3/4)

and irrational numbers, which cannot be expressed as a simple fraction, such as

• Square roots (e.g., \(\sqrt{2}\))
• Transcendental numbers (e.g., Ï€)

Real numbers are crucial in understanding intervals since they define the range of values within the intervals on the number line. The key to working with intervals is recognizing that they consist entirely of real numbers, either bounded or extending infinitely in one or both directions. In the given exercise, intervals like

• \((-\infty, -10]\)
• \((-\infty, -8]\)

cover a portion of real numbers extending from \(-\infty\) to specific values.

The union of intervals is a way to combine multiple intervals into one larger interval that encompasses all the values from the combined intervals. It's like merging two sets of numbers to include everything from both.
For example, if we have two intervals:

• \((1, 3)\)
• \((2, 5)\)

The union would encompass all numbers from 1 to 5, expressed as \((1, 5)\).

In the given exercise, we need to express the union of intervals \((-\infty, -10]\) and \((-\infty, -8]\).
The intervals overlap, so their union will simply be the broader of the two, i.e.,

• \((-\infty, -8]\)

This union contains all values that are in either interval. It's a compact way to represent all the real numbers that fit within both conditions at once.

Overlapping intervals occur when two intervals share some common values. These intervals will have a common range, often resulting in a single, larger interval when combined.
Two intervals may overlap completely or partially. For example:

• \((3, 7)\)
• \((5, 9)\)

Here, the overlap is from 5 to 7.

In the exercise, the intervals \((-\infty, -10]\) and \((-\infty, -8]\) overlap, since both start at \(-\infty\) and both cover parts of the number line ending at -10 and -8 respectively.
Analyzing the overlap helps in identifying the union, making it easier to see which interval fully contains the other without repeating any values.
The result, \((-\infty, -8]\), shows the importance of identifying overlaps to simplify the representation of intervals in set notation.