To solve this question, let's evaluate the set operations one by one. We'll match the correct sets to each operation based on the provided sets [tex]\( A, B, \)[/tex] and [tex]\( C \)[/tex].
Given:
- [tex]\( A = \{1, 2, 3, 4, 5\} \)[/tex]
- [tex]\( B = \{2, 4, 6\} \)[/tex]
- [tex]\( C = \{1, 3, 5\} \)[/tex]
Let's consider each operation step by step:
1. [tex]\( B \cap C \)[/tex]: This is the intersection of sets [tex]\( B \)[/tex] and [tex]\( C \)[/tex], which includes elements common to both sets.
- [tex]\( B \cap C = \{2, 4, 6\} \cap \{1, 3, 5\} = \emptyset \)[/tex]
(since there are no common elements).
So, [tex]\( B \cap C \)[/tex] matches the empty set (null set).
2. [tex]\( A \cap C \)[/tex]: This is the intersection of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], which includes elements common to both sets.
- [tex]\( A \cap C = \{1, 2, 3, 4, 5\} \cap \{1, 3, 5\} = \{1, 3, 5\} \)[/tex].
So, [tex]\( A \cap C \)[/tex] matches [tex]\(\{1, 3, 5\}\)[/tex].
3. [tex]\( A \cup C \)[/tex]: This is the union of sets [tex]\( A \)[/tex] and [tex]\( C \)[/tex], which includes all elements from both sets without repetition.
- [tex]\( A \cup C = \{1, 2, 3, 4, 5\} \cup \{1, 3, 5\} = \{1, 2, 3, 4, 5\} \)[/tex].
So, [tex]\( A \cup C \)[/tex] matches [tex]\( \{1, 2, 3, 4, 5\} \)[/tex].
4. [tex]\( A \cap B \)[/tex]: This is the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], which includes elements common to both sets.
- [tex]\( A \cap B = \{1, 2, 3, 4, 5\} \cap \{2, 4, 6\} = \{2, 4\} \)[/tex].
So, [tex]\( A \cap B \)[/tex] matches [tex]\(\{2, 4\}\)[/tex].
5. [tex]\( A \cup B \)[/tex]: This is the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], which includes all elements from both sets without repetition.
- [tex]\( A \cup B = \{1, 2, 3, 4, 5\} \cup \{2, 4, 6\} = \{1, 2, 3, 4, 5, 6\} \)[/tex].
So, [tex]\( A \cup B \)[/tex] matches [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex].
Final matching:
1. [tex]\( B \cap C \)[/tex] [tex]$\square$[/tex] [tex]\(\emptyset\)[/tex]
2. [tex]\( A \cap C \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 3, 5\}\)[/tex]
3. [tex]\( A \cup C \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 2, 3, 4, 5\}\)[/tex]
4. [tex]\( A \cap B \)[/tex] [tex]$\square$[/tex] [tex]\(\{2, 4\}\)[/tex]
5. [tex]\( A \cup B \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex]
Now, you can fill in the blanks based on this matching:
1. [tex]\( B \cap C \)[/tex] [tex]$\square$[/tex] null (empty set)
2. [tex]\( A \cap C \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 3, 5\}\)[/tex]
3. [tex]\( A \cup C \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 2, 3, 4, 5\}\)[/tex]
4. [tex]\( A \cap B \)[/tex] [tex]$\square$[/tex] [tex]\(\{2, 4\}\)[/tex]
5. [tex]\( A \cup B \)[/tex] [tex]$\square$[/tex] [tex]\(\{1, 2, 3, 4, 5, 6\}\)[/tex]
