Note: In general X ∪ Y = {a: a ϵ X or a ϵ Y}

X ∩ Y = {a:a ϵ X and a ϵ Y}.

i. A ∪ B = {x: x ϵ A or x ϵ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

ii. A ∪ C = {x: x ϵ A or x ϵ C}

= {1, 2, 3, 4, 5, 7, 8, 9, 10, 11}

iii. B ∪ C = {x: x ϵ B or x ϵ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

iv. B ∪ D = {x: x ϵ B or x ϵ D}

= {4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

v. A ∪ B = {x: x ϵ A or x ϵ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

A ∪ B ∪ C = {x: x ϵ A ∪ B or x ϵ C}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

vi. A ∪ B = {x: x ϵ A or x ϵ B}

= {1, 2, 3, 4, 5, 6, 7, 8}

A ∪ B ∪ D = {x: x ϵ A ∪ B or x ϵ D}

= {1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14}

vii. B ∪ C = {x: x ϵ B or x ϵ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

B ∪ C ∪ D = {x: x ϵ B ∪ C or x ϵ D}

= {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}

viii. B ∪ C = {x: x ϵ B or x ϵ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

A ∩ B ∪ C = {x:x ϵ A and x ϵ B ∪ C}.

= {4, 5}

ix. (A ∩ B) = {x:x ϵ A and x ϵ B}.

= {4, 5}

(B ∩ C) = {x:x ϵ B and x ϵ C}

= {7, 8}

(A ∩ B) ∩ (B ∩ C) = {x:x ϵ (A ∩ B) and x ϵ (B ∩ C)}.

= ϕ

x. A ∪ D = {x: x ϵ A or x ϵ D}

= {1, 2, 3, 4, 5, 10, 11, 12, 13, 14}.

B ∪ C = {x: x ϵ B or x ϵ C}

= {4, 5, 6, 7, 8, 9, 10, 11}

(A ∪ D) ∩ (B ∪ C) = {x:x ϵ (A ∪ D) and x ϵ (B ∪ C)}.

= {4, 5, 10, 11}.