Alright, let's solve the problem step-by-step.
Given sets:
- Universal set [tex]\( U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \)[/tex]
- Set [tex]\( A = \{1, 2, 4, 7\} \)[/tex]
- Set [tex]\( B = \{1, 3, 5, 7, 9\} \)[/tex]
We need to use De Morgan's Law to find [tex]\((A \cap B)^{\prime}\)[/tex].
### Step 1: Find [tex]\( A \cap B \)[/tex]
First, we find the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cap B = \{ x \mid x \in A \text{ and } x \in B \} \][/tex]
Elements in [tex]\( A \)[/tex] are: [tex]\( \{1, 2, 4, 7\} \)[/tex]
Elements in [tex]\( B \)[/tex] are: [tex]\( \{1, 3, 5, 7, 9\} \)[/tex]
The common elements in both sets are:
[tex]\[ A \cap B = \{1, 7\} \][/tex]
### Step 2: Find the complement of [tex]\( A \cap B \)[/tex]
The complement of [tex]\( A \cap B \)[/tex] with respect to the universal set [tex]\( U \)[/tex] is:
[tex]\[ (A \cap B)^{\prime} = U - (A \cap B) \][/tex]
[tex]\[ A \cap B = \{1, 7\} \][/tex]
So, the elements in the universal set [tex]\( U \)[/tex] that are not in [tex]\( A \cap B \)[/tex] are:
[tex]\[ (A \cap B)^{\prime} = \{2, 3, 4, 5, 6, 8, 9, 10\} \][/tex]
### Step 3: Verify Using De Morgan's Law
According to De Morgan's Law, [tex]\((A \cap B)^{\prime} = A^{\prime} \cup B^{\prime}\)[/tex].
First, find the complements of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A^{\prime} = U - A \][/tex]
[tex]\[ A = \{1, 2, 4, 7\} \][/tex]
[tex]\[ A^{\prime} = \{3, 5, 6, 8, 9, 10\} \][/tex]
[tex]\[ B^{\prime} = U - B \][/tex]
[tex]\[ B = \{1, 3, 5, 7, 9\} \][/tex]
[tex]\[ B^{\prime} = \{2, 4, 6, 8, 10\} \][/tex]
Then, find the union of [tex]\( A^{\prime} \)[/tex] and [tex]\( B^{\prime} \)[/tex]:
[tex]\[ A^{\prime} \cup B^{\prime} = \{3, 5, 6, 8, 9, 10\} \cup \{2, 4, 6, 8, 10\} \][/tex]
[tex]\[ A^{\prime} \cup B^{\prime} = \{2, 3, 4, 5, 6, 8, 9, 10\} \][/tex]
### Conclusion
As expected from De Morgan's Law:
[tex]\[ (A \cap B)^{\prime} = \{2, 3, 4, 5, 6, 8, 9, 10\} \][/tex]
which verifies our original statement.
Thus:
[tex]\[ (A \cap B)^{\prime} = A^{\prime} \cup B^{\prime} = \{2, 3, 4, 5, 6, 8, 9, 10\} \][/tex]
So, the correct answer is:
[tex]\[ (A \cap B)^{\prime} = \{ 2, 3, 4, 5, 6, 8, 9, 10 \} \][/tex]
