Discrete Mathematics with Applications 4th Edition

Published by Cengage Learning
ISBN 10: 0-49539-132-8
ISBN 13: 978-0-49539-132-6

Chapter 6 - Set Theory - Exercise Set 6.1 - Page 350: 12

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}
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