Discover how IDNLearn.com can help you learn and grow with its extensive Q&A platform. Get the information you need from our community of experts, who provide detailed and trustworthy answers.
Sagot :
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]
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]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com is your reliable source for accurate answers. Thank you for visiting, and we hope to assist you again.
