Let the universal set be ℝ, the set of all real numbers, and let A = {x ℝ | −3 ≤ x ≤ 0}, B = {x ℝ −1 < x < 2}, and C = {x ℝ | 5 < x ≤ 7}. Find each of the following: (a) A ∪ B x ℝ −3 ≤ x < 2 x ℝ −3 < x ≤ 2 x ℝ x ≤ −3 or x > 2 x ℝ x < −3 or x ≥ 2 ∅ (b) A ∩ B x ℝ −1 ≤ x < 0 x ℝ −1 < x ≤ 0 x ℝ x < −1 or x ≥ 0 x ℝ x ≤ −1 or x > 0 ∅ (c) Ac x ℝ −3 < x < 0 x ℝ −3 ≤ x ≤ 0 x ℝ x ≤ −3 or x ≥ 0 x ℝ x < −3 or x > 0 ∅

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let the universal set be ℝ, the set of all real numbers, and let 
A = {x  ℝ | −3 ≤ x ≤ 0},
 
B = {x  ℝ −1 < x < 2},
 and 
C = {x  ℝ | 5 < x ≤ 7}.
 Find each of the following:
(a)
A ∪ B
 
x  ℝ
 
−3 ≤ x < 2
 
 
x  ℝ
 
−3 < x ≤ 2
 
    
 
x  ℝ
 
x ≤ −3 or x > 2
 
 
x  ℝ
 
x < −3 or x ≥ 2
 
(b)
A ∩ B
 
x  ℝ
 
−1 ≤ x < 0
 
 
x  ℝ
 
−1 < x ≤ 0
 
    
 
x  ℝ
 
x < −1 or x ≥ 0
 
 
x  ℝ
 
x ≤ −1 or x > 0
 
(c)
Ac
 
x  ℝ
 
−3 < x < 0
 
 
x  ℝ
 
−3 ≤ x ≤ 0
 
    
 
x  ℝ
 
x ≤ −3 or x ≥ 0
 
 
x  ℝ
 
x < −3 or x > 0
 
(d)
A ∪ C
 
x  ℝ
 
−3 ≤ x ≤ 0 or 5 < x ≤ 7
 
 
x  ℝ
 
−3 < x < 0 or 5 < x ≤ 7
 
    
 
x  ℝ
 
−3 ≤ x ≤ 0 or 5 < x < 7
 
 
x  ℝ
 
−3 ≤ x ≤ 0 and 5 < x ≤ 7
 
(e)
A ∩ C
 
x  ℝ
 
−3 ≤ x ≤ 0 or 5 < x ≤ 7
 
 
x  ℝ
 
−3 < x < 0 or 5 < x ≤ 7
 
    
 
x  ℝ
 
−3 ≤ x ≤ 0 or 5 < x < 7
 
 
x  ℝ
 
−3 ≤ x ≤ 0 and 5 < x ≤ 7
 
(f)
Bc
 
x  ℝ
 
x < −1 or x > 2
 
 
x  ℝ
 
−1 ≤ x ≤ 2
 
    
 
x  ℝ
 
−1 < x < 2
 
 
x  ℝ
 
x ≤ −1 or x ≥ 2
 
 
x  ℝ
 
x ≤ −2 or x ≥ 1
 
(g)
Ac ∩ Bc
 
x  ℝ
 
−3 < x < 2
 
 
x  ℝ
 
x < −3 or x ≥ 2
 
    
 
x  ℝ
 
x ≤ −3 or x > 2
 
 
x  ℝ
 
−3 ≤ x ≤ 2
 
 
x  ℝ
 
x < −2 or x ≥ 3
 
(h)
Ac ∪ Bc
 
x  ℝ
 
x ≤ −1 or x > 0
 
 
x  ℝ
 
−1 ≤ x ≤ 0
 
    
 
x  ℝ
 
x < −1 or x ≥ 0
 
 
x  ℝ
 
−1 < x < 0
 
 
x  ℝ
 
x ≤ 0 or x > 1
 
(i)
(A ∩ B)c
 
x  ℝ
 
−1 ≤ x ≤ 0
 
 
x  ℝ
 
x ≤ 0 or x > 1
 
    
 
x  ℝ
 
x ≤ −1 or x > 0
 
 
x  ℝ
 
x < −1 or x ≥ 0
 
 
x  ℝ
 
−1 < x < 0
 
(j)
(A ∪ B)c
 
x  ℝ
 
−3 < x < 2
 
 
x  ℝ
 
x < −3 or x ≥ 2
 
    
 
x  ℝ
 
x ≤ −3 or x > 2
 
 
x  ℝ
 
−3 ≤ x ≤ 2
 
 
x  ℝ
 
x < −2 or x ≥ 3
 
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