To find the result for the given sets, we need to perform a couple of set operations. Let's go through the steps in detail.
Given the sets:
[tex]\[ U = \{28, 29, 30, 31, 32, 33, 34, 35, 36, 37\} \][/tex]
[tex]\[ A = \{31, 32, 33, 34\} \][/tex]
[tex]\[ B = \{28, 30, 32, 34, 36\} \][/tex]
[tex]\[ C = \{29, 31, 32, 36, 37\} \][/tex]
### Step 1: Find [tex]\( A \cup B \)[/tex] (Union of A and B)
The union of two sets [tex]\( A \cup B \)[/tex] contains all the elements that are in [tex]\( A \)[/tex], in [tex]\( B \)[/tex], or in both.
[tex]\[ A = \{31, 32, 33, 34\} \][/tex]
[tex]\[ B = \{28, 30, 32, 34, 36\} \][/tex]
[tex]\[ A \cup B = \{28, 30, 31, 32, 33, 34, 36\} \][/tex]
This union operation includes all distinct elements from both sets.
### Step 2: Find [tex]\( (A \cup B) \cap C \)[/tex] (Intersection of [tex]\( A \cup B \)[/tex] with C)
The intersection of two sets [tex]\( (A \cup B) \cap C \)[/tex] contains all the elements that are in both [tex]\( A \cup B \)[/tex] and [tex]\( C \)[/tex].
[tex]\[ A \cup B = \{28, 30, 31, 32, 33, 34, 36\} \][/tex]
[tex]\[ C = \{29, 31, 32, 36, 37\} \][/tex]
Now, identify the elements common to both [tex]\( A \cup B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ (A \cup B) \cap C = \{31, 32, 36\} \][/tex]
So, the steps yield the following results:
1. The union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is [tex]\( \{28, 30, 31, 32, 33, 34, 36\} \)[/tex].
2. The intersection of [tex]\( (A \cup B) \)[/tex] with [tex]\( C \)[/tex] is [tex]\( \{31, 32, 36\} \)[/tex].
Thus, the final sets we found are:
[tex]\[ A \cup B = \{28, 30, 31, 32, 33, 34, 36\} \][/tex]
[tex]\[ (A \cup B) \cap C = \{31, 32, 36\} \][/tex]
