Answer :
To solve the expression [tex]\( A' \cap \left(B \cup C'\right) \)[/tex], let’s follow these steps sequentially:
### Step 1: Calculate the Complement of [tex]\(A\)[/tex] ([tex]\(A'\)[/tex])
The complement of set [tex]\(A\)[/tex] (denoted [tex]\(A'\)[/tex]) includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(A\)[/tex].
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ U = \{1, 2, 3, 4, 5, 6\} \][/tex]
So,
[tex]\[ A' = U - A = \{5, 6\} \][/tex]
### Step 2: Calculate the Complement of [tex]\(C\)[/tex] ([tex]\(C'\)[/tex])
The complement of set [tex]\(C\)[/tex] (denoted [tex]\(C'\)[/tex]) includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(C\)[/tex].
[tex]\[ C = \{1, 2, 3, 4, 5\} \][/tex]
[tex]\[ U = \{1, 2, 3, 4, 5, 6\} \][/tex]
So,
[tex]\[ C' = U - C = \{6\} \][/tex]
### Step 3: Calculate the Union of [tex]\(B\)[/tex] and [tex]\(C'\)[/tex] ([tex]\(B \cup C'\)[/tex])
We now need to find the union of [tex]\(B\)[/tex] and [tex]\(C'\)[/tex], which includes all elements that are in [tex]\(B\)[/tex] or in [tex]\(C'\)[/tex] or in both sets.
[tex]\[ B = \{1, 3, 4\} \][/tex]
[tex]\[ C' = \{6\} \][/tex]
[tex]\[ B \cup C' = \{1, 3, 4\} \cup \{6\} = \{1, 3, 4, 6\} \][/tex]
### Step 4: Calculate the Intersection of [tex]\(A'\)[/tex] and [tex]\((B \cup C')\)[/tex]
Finally, we find the intersection of [tex]\(A'\)[/tex] and the union set [tex]\((B \cup C')\)[/tex]. The intersection includes all elements that are in both sets.
[tex]\[ A' = \{5, 6\} \][/tex]
[tex]\[ B \cup C' = \{1, 3, 4, 6\} \][/tex]
[tex]\[ A' \cap (B \cup C') = \{5, 6\} \cap \{1, 3, 4, 6\} = \{6\} \][/tex]
Therefore, the final result is:
[tex]\[ A' \cap (B \cup C') = \{6\} \][/tex]
Given the options, the correct answer is neither of the provided choices exactly. An appropriate response should state the explicit result.
### Correct Answer
[tex]\( A' \cap (B \cup C') = \{6\} \)[/tex]
### Step 1: Calculate the Complement of [tex]\(A\)[/tex] ([tex]\(A'\)[/tex])
The complement of set [tex]\(A\)[/tex] (denoted [tex]\(A'\)[/tex]) includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(A\)[/tex].
[tex]\[ A = \{1, 2, 3, 4\} \][/tex]
[tex]\[ U = \{1, 2, 3, 4, 5, 6\} \][/tex]
So,
[tex]\[ A' = U - A = \{5, 6\} \][/tex]
### Step 2: Calculate the Complement of [tex]\(C\)[/tex] ([tex]\(C'\)[/tex])
The complement of set [tex]\(C\)[/tex] (denoted [tex]\(C'\)[/tex]) includes all elements in the universal set [tex]\(U\)[/tex] that are not in [tex]\(C\)[/tex].
[tex]\[ C = \{1, 2, 3, 4, 5\} \][/tex]
[tex]\[ U = \{1, 2, 3, 4, 5, 6\} \][/tex]
So,
[tex]\[ C' = U - C = \{6\} \][/tex]
### Step 3: Calculate the Union of [tex]\(B\)[/tex] and [tex]\(C'\)[/tex] ([tex]\(B \cup C'\)[/tex])
We now need to find the union of [tex]\(B\)[/tex] and [tex]\(C'\)[/tex], which includes all elements that are in [tex]\(B\)[/tex] or in [tex]\(C'\)[/tex] or in both sets.
[tex]\[ B = \{1, 3, 4\} \][/tex]
[tex]\[ C' = \{6\} \][/tex]
[tex]\[ B \cup C' = \{1, 3, 4\} \cup \{6\} = \{1, 3, 4, 6\} \][/tex]
### Step 4: Calculate the Intersection of [tex]\(A'\)[/tex] and [tex]\((B \cup C')\)[/tex]
Finally, we find the intersection of [tex]\(A'\)[/tex] and the union set [tex]\((B \cup C')\)[/tex]. The intersection includes all elements that are in both sets.
[tex]\[ A' = \{5, 6\} \][/tex]
[tex]\[ B \cup C' = \{1, 3, 4, 6\} \][/tex]
[tex]\[ A' \cap (B \cup C') = \{5, 6\} \cap \{1, 3, 4, 6\} = \{6\} \][/tex]
Therefore, the final result is:
[tex]\[ A' \cap (B \cup C') = \{6\} \][/tex]
Given the options, the correct answer is neither of the provided choices exactly. An appropriate response should state the explicit result.
### Correct Answer
[tex]\( A' \cap (B \cup C') = \{6\} \)[/tex]
