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Sagot :
Let's solve this step by step.
Given the universal set and the subsets:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
[tex]\[ A = \{1, 2, 4, 7, 11\} \][/tex]
[tex]\[ B = \{1, 3, 4, 6, 7, 9, 11\} \][/tex]
[tex]\[ C = \{1, 5, 8, 11\} \][/tex]
### Step 1: Find [tex]\( A' \)[/tex] (the complement of [tex]\( A \)[/tex])
The complement of [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex] is given by the elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]:
[tex]\[ A' = U \setminus A = \{1, 2, 4, 7, 11\} = \{3, 5, 6, 8, 9, 10\} \][/tex]
### Step 2: Find [tex]\( A' \cup \emptyset \)[/tex]
The union of any set with the empty set is the set itself:
[tex]\[ A' \cup \emptyset = A' = \{3, 5, 6, 8, 9, 10\} \][/tex]
### Step 3: Find [tex]\( B \cap C \)[/tex]
The intersection of [tex]\( B \)[/tex] and [tex]\( C \)[/tex] is the set of elements that are in both [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ B \cap C = \{1, 3, 4, 6, 7, 9, 11\} \cap \{1, 5, 8, 11\} = \{1, 11\} \][/tex]
### Step 4: Find [tex]\( A' \cup B \)[/tex]
The union of [tex]\( A' \)[/tex] and [tex]\( B \)[/tex] includes all elements that are in either [tex]\( A' \)[/tex] or [tex]\( B \)[/tex]:
[tex]\[ A' \cup B = \{3, 5, 6, 8, 9, 10\} \cup \{1, 3, 4, 6, 7, 9, 11\} = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
### Step 5: Find [tex]\( (C \cap B) \cap A' \)[/tex]
First, find the intersection of [tex]\( C \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ C \cap B = \{1, 5, 8, 11\} \cap \{1, 3, 4, 6, 7, 9, 11\} = \{1, 11\} \][/tex]
Next, find the intersection of this result with [tex]\( A' \)[/tex]:
[tex]\[ (C \cap B) \cap A' = \{1, 11\} \cap \{3, 5, 6, 8, 9, 10\} = \emptyset \][/tex]
So, the final answers are:
1. [tex]\( A' \cup \emptyset = \{3, 5, 6, 8, 9, 10\} \)[/tex]
2. [tex]\( B \cap C = \{1, 11\} \)[/tex]
3. [tex]\( A' \cup B = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \)[/tex]
4. [tex]\( (C \cap B) \cap A' = \emptyset \)[/tex]
Here they are, arranged with appropriate formatting:
[tex]\[ A' \cup \emptyset = \{3, 5, 6, 8, 9, 10\} \][/tex]
[tex]\[ B \cap C = \{1, 11\} \][/tex]
[tex]\[ A' \cup B = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
[tex]\[ (C \cap B) \cap A' = \emptyset \][/tex]
Given the universal set and the subsets:
[tex]\[ U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
[tex]\[ A = \{1, 2, 4, 7, 11\} \][/tex]
[tex]\[ B = \{1, 3, 4, 6, 7, 9, 11\} \][/tex]
[tex]\[ C = \{1, 5, 8, 11\} \][/tex]
### Step 1: Find [tex]\( A' \)[/tex] (the complement of [tex]\( A \)[/tex])
The complement of [tex]\( A \)[/tex] in the universal set [tex]\( U \)[/tex] is given by the elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]:
[tex]\[ A' = U \setminus A = \{1, 2, 4, 7, 11\} = \{3, 5, 6, 8, 9, 10\} \][/tex]
### Step 2: Find [tex]\( A' \cup \emptyset \)[/tex]
The union of any set with the empty set is the set itself:
[tex]\[ A' \cup \emptyset = A' = \{3, 5, 6, 8, 9, 10\} \][/tex]
### Step 3: Find [tex]\( B \cap C \)[/tex]
The intersection of [tex]\( B \)[/tex] and [tex]\( C \)[/tex] is the set of elements that are in both [tex]\( B \)[/tex] and [tex]\( C \)[/tex]:
[tex]\[ B \cap C = \{1, 3, 4, 6, 7, 9, 11\} \cap \{1, 5, 8, 11\} = \{1, 11\} \][/tex]
### Step 4: Find [tex]\( A' \cup B \)[/tex]
The union of [tex]\( A' \)[/tex] and [tex]\( B \)[/tex] includes all elements that are in either [tex]\( A' \)[/tex] or [tex]\( B \)[/tex]:
[tex]\[ A' \cup B = \{3, 5, 6, 8, 9, 10\} \cup \{1, 3, 4, 6, 7, 9, 11\} = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
### Step 5: Find [tex]\( (C \cap B) \cap A' \)[/tex]
First, find the intersection of [tex]\( C \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ C \cap B = \{1, 5, 8, 11\} \cap \{1, 3, 4, 6, 7, 9, 11\} = \{1, 11\} \][/tex]
Next, find the intersection of this result with [tex]\( A' \)[/tex]:
[tex]\[ (C \cap B) \cap A' = \{1, 11\} \cap \{3, 5, 6, 8, 9, 10\} = \emptyset \][/tex]
So, the final answers are:
1. [tex]\( A' \cup \emptyset = \{3, 5, 6, 8, 9, 10\} \)[/tex]
2. [tex]\( B \cap C = \{1, 11\} \)[/tex]
3. [tex]\( A' \cup B = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \)[/tex]
4. [tex]\( (C \cap B) \cap A' = \emptyset \)[/tex]
Here they are, arranged with appropriate formatting:
[tex]\[ A' \cup \emptyset = \{3, 5, 6, 8, 9, 10\} \][/tex]
[tex]\[ B \cap C = \{1, 11\} \][/tex]
[tex]\[ A' \cup B = \{1, 3, 4, 5, 6, 7, 8, 9, 10, 11\} \][/tex]
[tex]\[ (C \cap B) \cap A' = \emptyset \][/tex]
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