Answer
A = {x ϵ R | -3 ≤ x ≤ 0}
B = {x ϵ R | -1 < x < 2}
C = {x ϵ R | 6 < x ≤ 8}
a. A⋃B = {x ϵ R | -3 ≤ x < 2}
b. A⋂B = {x ϵ R | -1 < x ≤ 0}
c. Ac = {x ϵ R | x < -3 or 0 < x}
d. A⋃C = {x ϵ R | -3 ≤ x ≤ 0 or 6 < x ≤ 8}
e. A⋂C = ∅
f. Bc = {x ϵ R | x ≤ -1 or 2 ≤ x}
g. Ac⋂Bc = {x ϵ R | x < -3 or 2 ≤ x}
h. Ac⋃Bc = {x ϵ R | x ≤ -1 or 0 < x}
i. (A⋂B)c = {x ϵ R | x ≤ -1 or 0 < x}
j. (A⋃B)c = {x ϵ R | x < -3 or 2 ≤ x}
Work Step by Step
A = {x ϵ R | -3 ≤ x ≤ 0}
B = {x ϵ R | -1 < x < 2}
C = {x ϵ R | 6 < x ≤ 8}
a. A⋃B = {x ϵ R | x is in at least one of the intervals [-3, 0] or (-1, 2)}
= {x ϵ R | -3 ≤ x < 2}
b. A⋂B = {x ϵ R | x is in all of the intervals [-3, 0] or (-1, 2)}
= {x ϵ R | -1 < x ≤ 0}
c. Ac = {x ϵ R | x is in all but the interval [-3, 0]}
= {x ϵ R | x < -3 or 0 < x}
d. A⋃C = {x ϵ R | x is at least one of the intervals [-3, 0] or (6, 8]}
= {x ϵ R | -3 ≤ x ≤ 0 or 6 < x ≤ 8}
e. A⋂C = {x ϵ R | x is all of the intervals [-3, 0] and (6, 8]}
= ∅
f. Bc = {x ϵ R | x is all but the interval (-1, 2)}
= {x ϵ R | x ≤ -1 or 2 ≤ x}
g. Ac⋂Bc {x ϵ R | x is in all of the intervals “x < -3 or 0 < x” and “x ≤ -1 or 2 ≤ x”}
= {x ϵ R | x < -3 or 2 ≤ x}
h. Ac⋃Bc = {x ϵ R | x is at least one of the intervals “x < -3 or 0 < x” or “x ≤ -1 or 2 ≤ x”}
= {x ϵ R | x ≤ -1 or 0 < x}
i. (A⋂B)c = {x ϵ R | x is in all but the interval -1 < x ≤ 0}
= {x ϵ R | x ≤ -1 or 0 < x}
j. (A⋃B)c = {x ϵ R | x is in all but the interval -3 ≤ x < 2}
= {x ϵ R | x < -3 or 2 ≤ x}