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Problem 6. Prove or disprove the following statements: 1. Let A, B and C be sets. Then A ? (B ? C) = (A ? B) ? (A ? C). 2. If A and B are sets, then P(A) ? P(B) = P(A ? B). 3. If A and B are sets, then P(A) ? P(B) = P(A ? B). 4. If A, B and C are sets, then A – (B ? C) = (A – B) ? (A – C). 5. Suppose A, B and C are sets. If A = B – C, then B = A ? C. 6. Let A, B, C, D be sets, then (A × B) ? (C × D) = (A ? C) × (B ? D). 7. If m, n ? ?, then {x ? ? : mn | x} ? {x ? ? : m | x} ? {x ? ? : n | x}. 8. If m, n ? ?, then {x ? ? : mn | x} = {x ? ? : m | x} ? {x ? ? : n | x}.

          Problem 6. Prove or disprove the following statements:

1. Let A, B and C be sets. Then A ? (B ? C) = (A ? B) ? (A ? C).
2. If A and B are sets, then P(A) ? P(B) = P(A ? B).
3. If A and B are sets, then P(A) ? P(B) = P(A ? B).
4. If A, B and C are sets, then A – (B ? C) = (A – B) ? (A – C).
5. Suppose A, B and C are sets. If A = B – C, then B = A ? C.
6. Let A, B, C, D be sets, then (A × B) ? (C × D) = (A ? C) × (B ? D).
7. If m, n ? ?, then {x ? ? : mn | x} ? {x ? ? : m | x} ? {x ? ? : n | x}.
8. If m, n ? ?, then {x ? ? : mn | x} = {x ? ? : m | x} ? {x ? ? : n | x}.

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Problem 6. Prove or disprove the following statements:

1. Let A, B and C be sets. Then A ? (B ? C) = (A ? B) ? (A ? C).
2. If A and B are sets, then P(A) ? P(B) = P(A ? B).
3. If A and B are sets, then P(A) ? P(B) = P(A ? B).
4. If A, B and C are sets, then A – (B ? C) = (A – B) ? (A – C).
5. Suppose A, B and C are sets. If A = B – C, then B = A ? C.
6. Let A, B, C, D be sets, then (A × B) ? (C × D) = (A ? C) × (B ? D).
7. If m, n ? ?, then x ? ? : mn | x ? x ? ? : m | x ? x ? ? : n | x.
8. If m, n ? ?, then x ? ? : mn | x = x ? ? : m | x ? x ? ? : n | x.

Added by James H.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Adi S

09/13/2023

Step 1

Let A, B, and C be sets. Then A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Proof: Let x be an element in A ∩ (B ∪ C). Then x ∈ A and x ∈ (B ∪ C). So, x ∈ A and (x ∈ B or x ∈ C). This means (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C). Thus, x ∈ (A ∩ B) ∪ (A ∩ C). Conversely, let x Show more…

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Problem 6. Prove or disprove the following statements: 1. Let A, B and C be sets. Then A ∩ (B ∡ C) = (A ∩ B) ∡ (A ∩ C). 2. If A and B are sets, then P(A) ∩ P(B) = P(A ∩ B). 3. If A and B are sets, then P(A) ∡ P(B) = P(A ∡ B). 4. If A, B and C are sets, then A – (B ∡ C) = (A – B) ∡ (A – C). 5. Suppose A, B and C are sets. If A = B – C, then B = A ∡ C. 6. Let A, B, C, D be sets, then (A × B) ∡ (C × D) = (A ∡ C) × (B ∡ D). 7. If m, n ∈ ℤ, then {x ∈ ℤ : mn | x} ⊆ {x ∈ ℤ : m | x} ∩ {x ∈ ℤ : n | x}. 8. If m, n ∈ ℤ, then {x ∈ ℤ : mn | x} = {x ∈ ℤ : m | x} ∩ {x ∈ ℤ : n | x}.

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