Sure, I'll guide you through understanding the union of two sets, [tex]\( A \)[/tex] and [tex]\( B \)[/tex], resulting in [tex]\( B \cup A = (-4, -3, -2, 0, 1, 2, 4, 5, 9) \)[/tex].
### Step-by-Step Explanation
1. Understand the Union Operation:
- The union of two sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] denoted as [tex]\( B \cup A \)[/tex], is the set containing all the elements that are in [tex]\( A \)[/tex], in [tex]\( B \)[/tex], or in both.
- Essentially, [tex]\( B \cup A \)[/tex] combines all unique elements from sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
2. Listing Elements:
- According to the given result, the elements that form [tex]\( B \cup A \)[/tex] are:
[tex]\[
B \cup A = \{ -4, -3, -2, 0, 1, 2, 4, 5, 9 \}
\][/tex]
3. Analyzing the Result Set:
- Notice [tex]\( B \cup A \)[/tex] is a set that includes:
[tex]\[
\text{Negative Numbers:} -4, -3, -2
\][/tex]
[tex]\[
\text{Non-negative Numbers:} 0, 1, 2, 4, 5, 9
\][/tex]
4. Forming Possible Sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- The sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] could take various forms as long as their union results in the set [tex]\( B \cup A \)[/tex].
- Let's denote the sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] in such a way that their union leads to the given numbers.
### Possible Example Sets
Here is one example of what the sets could be:
- Set [tex]\( A \)[/tex]:
[tex]\[
A = \{ -4, -2, 1, 5 \}
\][/tex]
- Set [tex]\( B \)[/tex]:
[tex]\[
B = \{ -3, 0, 2, 4, 9 \}
\][/tex]
Now, let's check if the union of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] includes all unique elements from the given result:
- Combine [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
A \cup B = \{ -4, -3, -2, 0, 1, 2, 4, 5, 9 \}
\][/tex]
This combination covers all elements of the given result set. The exact sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] could be different, and there is not necessarily only one solution.
### Conclusion
In summary, the operation [tex]\( B \cup A = (-4, -3, -2, 0, 1, 2, 4, 5, 9) \)[/tex] states that the union of sets [tex]\( B \)[/tex] and [tex]\( A \)[/tex] results in the set containing those specific elements. Various combinations of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] can lead to this result as long as their union covers all the provided elements without repetition.
