To find the union of the sets [tex]\( \bigcup_{t=1}^3 A_t \)[/tex], we need to combine all the elements from the sets [tex]\( A_1 \)[/tex], [tex]\( A_2 \)[/tex], and [tex]\( A_3 \)[/tex].
Let's denote the sets as follows:
- [tex]\( A_1 = \{1, 2, 3\} \)[/tex]
- [tex]\( A_2 = \{3, 4, 5\} \)[/tex]
- [tex]\( A_3 = \{5, 6, 7\} \)[/tex]
The union of sets [tex]\( A_1 \)[/tex], [tex]\( A_2 \)[/tex], and [tex]\( A_3 \)[/tex] is a set that contains all the unique elements from each of these sets.
Step-by-step:
1. Start with [tex]\( A_1 \)[/tex]:
[tex]\[
A_1 = \{1, 2, 3\}
\][/tex]
2. Add the elements of [tex]\( A_2 \)[/tex] to the union:
[tex]\[
A_2 = \{3, 4, 5\}
\][/tex]
Combine [tex]\( A_1 \)[/tex] and [tex]\( A_2 \)[/tex], resulting in [tex]\( \{1, 2, 3, 4, 5\} \)[/tex]. Notice that [tex]\( 3 \)[/tex] is already included, so we do not repeat it.
3. Add the elements of [tex]\( A_3 \)[/tex] to the union:
[tex]\[
A_3 = \{5, 6, 7\}
\][/tex]
Combine this with the previous result [tex]\( \{1, 2, 3, 4, 5\} \)[/tex], resulting in [tex]\( \{1, 2, 3, 4, 5, 6, 7\} \)[/tex]. Elements [tex]\( 5 \)[/tex] are already included, so we do not repeat them.
So, the union of [tex]\( A_1 \)[/tex], [tex]\( A_2 \)[/tex], and [tex]\( A_3 \)[/tex] is:
[tex]\[ \bigcup_{t=1}^3 A_t = \{1, 2, 3, 4, 5, 6, 7\} \][/tex]
This is the final set that contains all the unique elements from [tex]\( A_1 \)[/tex], [tex]\( A_2 \)[/tex], and [tex]\( A_3 \)[/tex].
