Certainly! Let's analyze the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] and determine their intersection [tex]\( A \cap B \)[/tex] and union [tex]\( A \cup B \)[/tex] using interval notation.
Step 1: Define the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] using the given conditions.
- Set [tex]\( A \)[/tex] is defined by [tex]\( A = \{ v \mid v \geq 1 \} \)[/tex]. This means [tex]\( A \)[/tex] contains all real numbers [tex]\( v \)[/tex] that are greater than or equal to 1. In interval notation, we can express this as:
[tex]\[
A = [1, \infty)
\][/tex]
- Set [tex]\( B \)[/tex] is defined by [tex]\( B = \{ v \mid v < 8 \} \)[/tex]. This means [tex]\( B \)[/tex] contains all real numbers [tex]\( v \)[/tex] that are less than 8. In interval notation, we can express this as:
[tex]\[
B = (-\infty, 8)
\][/tex]
Step 2: Find the intersection [tex]\( A \cap B \)[/tex].
- The intersection [tex]\( A \cap B \)[/tex] consists of elements that are in both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. We need to identify the values of [tex]\( v \)[/tex] that satisfy both conditions [tex]\( v \geq 1 \)[/tex] and [tex]\( v < 8 \)[/tex].
- Only numbers that are both greater than or equal to 1 and less than 8 will be in the intersection.
Combining these conditions, the set [tex]\( A \cap B \)[/tex] can be written in interval notation as:
[tex]\[
A \cap B = [1, 8)
\][/tex]
Step 3: Find the union [tex]\( A \cup B \)[/tex].
- The union [tex]\( A \cup B \)[/tex] consists of elements that are in either set [tex]\( A \)[/tex] or set [tex]\( B \)[/tex] or in both. We need to combine the intervals that encompass all values in [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
- Set [tex]\( A \)[/tex] includes all elements from 1 to infinity, and set [tex]\( B \)[/tex] includes all elements from negative infinity to 8.
Since [tex]\( A \)[/tex] already includes everything from 1 onward, and [tex]\( B \)[/tex] includes everything less than 8, the union of these two sets will cover all real numbers starting from 1 to beyond.
Combining these intervals, the set [tex]\( A \cup B \)[/tex] can be written in interval notation as:
[tex]\[
A \cup B = [1, \infty)
\][/tex]
Summary:
- The intersection of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[
A \cap B = [1, 8)
\][/tex]
- The union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[
A \cup B = [1, \infty)
\][/tex]
