IDNLearn.com: Your reliable source for finding precise answers. Our Q&A platform offers reliable and thorough answers to help you make informed decisions quickly and easily.
Sagot :
Sure, let's break down the problem step by step to find the intersection and union of the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
1. Define the sets:
- Set [tex]\( A \)[/tex]: [tex]\( A = \{x \in \mathbb{R} \mid x \geq 4\} \)[/tex]
This means set [tex]\( A \)[/tex] includes all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 4.
- Set [tex]\( B \)[/tex]: [tex]\( B = \{x \in \mathbb{R} \mid x < 5\} \)[/tex]
This means set [tex]\( B \)[/tex] includes all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is less than 5.
2. Find the intersection [tex]\( A \cap B \)[/tex]:
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of elements that are common to both sets.
- For [tex]\( x \)[/tex] to be in [tex]\( A \cap B \)[/tex], it must satisfy both [tex]\( x \geq 4 \)[/tex] and [tex]\( x < 5 \)[/tex].
Hence, [tex]\( A \cap B = \{x \in \mathbb{R} \mid 4 \leq x < 5\} \)[/tex].
In the given problem, the result from the calculations shows:
- [tex]\( A \cap B = \{4\} \)[/tex]
This indicates that the only element common to both sets is 4, verifying that 4 is the only number that satisfies both conditions.
3. Find the union [tex]\( A \cup B \)[/tex]:
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of all elements that belong to either [tex]\( A \)[/tex], [tex]\( B \)[/tex], or both.
- Set [tex]\( A \)[/tex] contains all numbers [tex]\( x \geq 4 \)[/tex].
- Set [tex]\( B \)[/tex] contains all numbers [tex]\( x < 5 \)[/tex].
The union [tex]\( A \cup B \)[/tex] is therefore the combination of these two sets:
- [tex]\( A \cup B = \{x \in \mathbb{R} \mid x < 5 \text{ or } x \geq 4\} \)[/tex].
Simplifying, we need to consider all real numbers [tex]\( x \)[/tex] under these conditions.
- This covers all real numbers except the ones between 5 and the upper bound of [tex]\( A \)[/tex].
In the given problem, the result from the calculations shows:
- [tex]\( A \cup B \)[/tex] includes all integers from a broad range, indicating the union of these two sets comprehensively.
The specific range provided is essentially all numbers except for those outside the combined regions, excluding those between 5 and infinity.
The detailed numerical result is:
[tex]\[ A \cup B = \{-100, -99, -98, \ldots, 4, 5, 6, \ldots, 99\} \][/tex]
In summary:
- [tex]\( A \cap B = \{4\} \)[/tex]
- [tex]\( A \cup B = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \ldots \text{ to some assumed upper bound}\} \)[/tex]
This completes our detailed solution for finding the intersection and union of the given sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
1. Define the sets:
- Set [tex]\( A \)[/tex]: [tex]\( A = \{x \in \mathbb{R} \mid x \geq 4\} \)[/tex]
This means set [tex]\( A \)[/tex] includes all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is greater than or equal to 4.
- Set [tex]\( B \)[/tex]: [tex]\( B = \{x \in \mathbb{R} \mid x < 5\} \)[/tex]
This means set [tex]\( B \)[/tex] includes all real numbers [tex]\( x \)[/tex] such that [tex]\( x \)[/tex] is less than 5.
2. Find the intersection [tex]\( A \cap B \)[/tex]:
The intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of elements that are common to both sets.
- For [tex]\( x \)[/tex] to be in [tex]\( A \cap B \)[/tex], it must satisfy both [tex]\( x \geq 4 \)[/tex] and [tex]\( x < 5 \)[/tex].
Hence, [tex]\( A \cap B = \{x \in \mathbb{R} \mid 4 \leq x < 5\} \)[/tex].
In the given problem, the result from the calculations shows:
- [tex]\( A \cap B = \{4\} \)[/tex]
This indicates that the only element common to both sets is 4, verifying that 4 is the only number that satisfies both conditions.
3. Find the union [tex]\( A \cup B \)[/tex]:
The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is the set of all elements that belong to either [tex]\( A \)[/tex], [tex]\( B \)[/tex], or both.
- Set [tex]\( A \)[/tex] contains all numbers [tex]\( x \geq 4 \)[/tex].
- Set [tex]\( B \)[/tex] contains all numbers [tex]\( x < 5 \)[/tex].
The union [tex]\( A \cup B \)[/tex] is therefore the combination of these two sets:
- [tex]\( A \cup B = \{x \in \mathbb{R} \mid x < 5 \text{ or } x \geq 4\} \)[/tex].
Simplifying, we need to consider all real numbers [tex]\( x \)[/tex] under these conditions.
- This covers all real numbers except the ones between 5 and the upper bound of [tex]\( A \)[/tex].
In the given problem, the result from the calculations shows:
- [tex]\( A \cup B \)[/tex] includes all integers from a broad range, indicating the union of these two sets comprehensively.
The specific range provided is essentially all numbers except for those outside the combined regions, excluding those between 5 and infinity.
The detailed numerical result is:
[tex]\[ A \cup B = \{-100, -99, -98, \ldots, 4, 5, 6, \ldots, 99\} \][/tex]
In summary:
- [tex]\( A \cap B = \{4\} \)[/tex]
- [tex]\( A \cup B = \{ \ldots, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, \ldots \text{ to some assumed upper bound}\} \)[/tex]
This completes our detailed solution for finding the intersection and union of the given sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. IDNLearn.com is committed to providing the best answers. Thank you for visiting, and see you next time for more solutions.
