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Let [tex]\( U = \{1, 2, 3, \ldots, 6\} \)[/tex], [tex]\( A = \{1, 2, 3, 4\} \)[/tex], [tex]\( B = \{1, 3, 4\} \)[/tex], and [tex]\( C = \{1, 2, 3, 4, 5\} \)[/tex].

Perform the indicated operations:
[tex]\[ A' \cap (B \cup C') \][/tex]

Select the correct choice below and, if necessary, fill in the answer box to complete your choice:

A. [tex]\( A' \cap (B \cup C') = \{6, 7\} \)[/tex] [tex]\( \boxed{} \)[/tex]
(Use ascending order. Use a comma to separate answers as needed.)

B. The solution is [tex]\( \varnothing \)[/tex].

Sagot :

GinnyAnswer
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]
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