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].
