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Step-by-Step Solution
Step 1: Understand the Definitions of Sets A, B, and C
• Set A is defined by the condition |x - 2| > 1 .
This inequality holds if:
(1) x - 2 > 1 \implies x > 3 , or
(2) x - 2 < -1 \implies x < 1 .
Hence,
A = (-\infty,\,1) \,\cup\, (3,\,\infty).
• Set B is defined by the condition \sqrt{x^2 - 3} > 1 .
This means
x^2 - 3 > 1 \quad \Longrightarrow \quad x^2 > 4 \quad \Longrightarrow \quad |x| > 2.
Therefore, x < -2 or x > 2 . Thus,
B = (-\infty,\,-2) \,\cup\, (2,\,\infty).
• Set C is defined by the condition |x - 4| \ge 2 .
This inequality holds if:
(1) x - 4 \ge 2 \implies x \ge 6 , or
(2) x - 4 \le -2 \implies x \le 2 .
Hence,
C = (-\infty,\,2] \,\cup\, [6,\,\infty).
Step 2: Find the Intersection A ∩ B ∩ C
First, note:
A = (-\infty,\,1) \cup (3,\,\infty), \quad
B = (-\infty,\,-2) \cup (2,\,\infty), \quad
C = (-\infty,\,2] \cup [6,\,\infty).
We can combine them systematically:
B ∩ C:
(-\infty,\,-2) \cap (-\infty,\,2] = (-\infty,\,-2)
(-\infty,\,-2) \cap [6,\,\infty) = \varnothing
(2,\,\infty) \cap (-\infty,\,2] = \varnothing
(2,\,\infty) \cap [6,\,\infty) = [6,\,\infty)
So,
B \cap C = (-\infty,\,-2) \cup [6,\,\infty).
A ∩ (B ∩ C):
(-\infty,\,1) \cap (-\infty,\,-2) = (-\infty,\,-2)
(-\infty,\,1) \cap [6,\,\infty) = \varnothing
(3,\,\infty) \cap (-\infty,\,-2) = \varnothing
(3,\,\infty) \cap [6,\,\infty) = [6,\,\infty)
Hence,
A \cap B \cap C = (-\infty,\,-2) \cup [6,\,\infty).
Step 3: Determine (A ∩ B ∩ C)c
We have
A \cap B \cap C = (-\infty,\,-2) \,\cup\, [6,\,\infty).
Its complement (in \mathbb{R} ) is all real numbers not in that union:
(A \cap B \cap C)^c
= \mathbb{R} \setminus \bigl((-\infty,\,-2) \,\cup\, [6,\,\infty)\bigr)
= [-2,\,6).
Step 4: Find the Integers in (A ∩ B ∩ C)c
The set of integers Z is all integers from -\infty to \infty . We want:
Z \cap (A \cap B \cap C)^c
= Z \cap [-2,\,6).
The integers in [-2,\,6) are:
\{-2,\,-1,\,0,\,1,\,2,\,3,\,4,\,5\}.
There are 8 integers total.
Step 5: Count the Number of Subsets
The number of distinct subsets of a set with n elements is 2^n . Here, n=8 , so
\text{Number of subsets} = 2^8 = 256.
Answer
The number of subsets is 256.
