Exercises on sets

1. Prove that A ∪ (B ∪ C) = (A ∪ B) ∪ C (associative laws for union of sets)

2. Prove that (A ∩ B) ∩ C = A ∩ (B ∩ C) (associative laws for intersections of sets)

3. Prove that (A ∪ B) = A only if B ⊆ A.

4. Prove the following distributive properties

   • (A ∪ B) ∩C = (A ∩ C) ∪ (B ∩ C) (distributive of the intersection with regards to union)

   • (A ∩ B) ∪C = (A ∪ C) ∩ (B ∪ C) (distributive of the union with regards to intersection)

5. Prove De Morgan's laws

   • 𝓒(A ∩ B) = 𝓒A ∪ 𝓒B

   • 𝓒(A ∪ B) = 𝓒A ∩ 𝓒B

6. Prove that for any sets A, B, C:

   (A − B) U (B − A) = (A U B) − (A n B)

Solutions

1. To prove this identity it necessary to show that any element in the left-hand side is also an element of the right-hand side. We have

   x ∈ A ∪ (B ∪ C) ⇒ x ∈ A or x ∈ B ∪ C ⇒ x ∈ A or x ∈ B or x ∈ C ⇒ x ∈ A ∪ B or x ∈ C ⇒ x ∈ (A ∪ B) ∪ C.

   Analogously by inverting the order of the implications we prove that

   x ∈ (A ∪ B) ∪ C ⇒ x ∈ A ∪ (B ∪ C)

   it is thus correct to remove parenthesis and write

   A ∪ B ∪ C = (A ∪ B) ∪ C = A ∪ (B ∪ C) ■

2. For the sake of simplicity let

   D = A ∩ B and E = B ∩ C, then

   D = {x: x ∈ A and x ∈ B}

   E = {x: x ∈ B and x ∈ C}

   thus

   (A ∩ B) ∩ C = D ∩ C = {x: x ∈ D and x ∈ C} = {x: x ∈ A and x ∈ B and x ∈ C}

   hence the equality. ■

3. [A ∪ B = A] ⇒ [x ∈ A ∪ B ⇐⇒ x ∈ A] ⇒ B ⊆ A.  ■

4. We have

   • x ∈ (A ∪ B) ∩ C ⇒ x ∈ C and x ∈ A ∪ B ⇒ [x ∈ C and [x ∈ A or x ∈ B]] ⇒ [x ∈ C and x ∈ A] or [x ∈ C and x ∈ B] ⇒ x ∈ (C ∩ A) ∪ (C ∩ B).

   • x ∈ (A ∩ B) ∪ C ⇒ x ∈ C and x ∈ A ∩ B ⇒ [x ∈ C and [x ∈ A and x ∈ B]] ⇒ [x ∈ C and x ∈ A] and [x ∈ C and x ∈ B] ⇒ x ∈ (C ∩ A) ∪ (C ∩ B). ■□

5. We have

   • x ∈ 𝓒A ∪ 𝓒B ⇒ x ∈ 𝓒A or x ∈ 𝓒B ⇒ x ∉ A ∩ B ⇒ x ∈ 𝓒(A ∩ B)

   • x ∈ 𝓒A∩ 𝓒B ⇒ x ∈ 𝓒A and x ∈ 𝓒B ⇒ x ∉ A ∪ B ⇒ x ∈ 𝓒(A ∩ B) ■

6. (A − B) U (B − A) = {x: x ∈ A | x ∉ B} U {x: x ∈ B | x ∉ A} = {x: x ∈ A, x ∈ B | x ∉ A, x ∉ B} = (A U B) − (A n B). ■

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