MAT3702 Assignment 1 solutions 2026
DUE: 13 MAY 2026
ALL QUESTIONS ARE ANSWERED FULLY AND VERIFIED
, Assignment 1
1. Prove:
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Proof (element method):
Let 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶)
Then:
• Either 𝑥 ∈ 𝐴, OR
• 𝑥 ∈ 𝐵 ∩ 𝐶 ⇒ 𝑥 ∈ 𝐵AND 𝑥 ∈ 𝐶
So:
• If 𝑥 ∈ 𝐴, then 𝑥 ∈ 𝐴 ∪ 𝐵AND 𝑥 ∈ 𝐴 ∪ 𝐶
• If 𝑥 ∈ 𝐵 ∩ 𝐶, then 𝑥 ∈ 𝐵 ⇒ 𝑥 ∈ 𝐴 ∪ 𝐵and 𝑥 ∈ 𝐶 ⇒ 𝑥 ∈ 𝐴 ∪ 𝐶
Thus:
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
So:
𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Now reverse:
Let 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Then:
• 𝑥 ∈ 𝐴 ∪ 𝐵 ⇒ 𝑥 ∈ 𝐴OR 𝑥 ∈ 𝐵
• 𝑥 ∈ 𝐴 ∪ 𝐶 ⇒ 𝑥 ∈ 𝐴OR 𝑥 ∈ 𝐶
Cases:
• If 𝑥 ∈ 𝐴→ done
• If not in A, then must be in both B and C → 𝑥 ∈ 𝐵 ∩ 𝐶
Thus:
𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶)
Proven.
DUE: 13 MAY 2026
ALL QUESTIONS ARE ANSWERED FULLY AND VERIFIED
, Assignment 1
1. Prove:
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Proof (element method):
Let 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶)
Then:
• Either 𝑥 ∈ 𝐴, OR
• 𝑥 ∈ 𝐵 ∩ 𝐶 ⇒ 𝑥 ∈ 𝐵AND 𝑥 ∈ 𝐶
So:
• If 𝑥 ∈ 𝐴, then 𝑥 ∈ 𝐴 ∪ 𝐵AND 𝑥 ∈ 𝐴 ∪ 𝐶
• If 𝑥 ∈ 𝐵 ∩ 𝐶, then 𝑥 ∈ 𝐵 ⇒ 𝑥 ∈ 𝐴 ∪ 𝐵and 𝑥 ∈ 𝐶 ⇒ 𝑥 ∈ 𝐴 ∪ 𝐶
Thus:
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
So:
𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Now reverse:
Let 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Then:
• 𝑥 ∈ 𝐴 ∪ 𝐵 ⇒ 𝑥 ∈ 𝐴OR 𝑥 ∈ 𝐵
• 𝑥 ∈ 𝐴 ∪ 𝐶 ⇒ 𝑥 ∈ 𝐴OR 𝑥 ∈ 𝐶
Cases:
• If 𝑥 ∈ 𝐴→ done
• If not in A, then must be in both B and C → 𝑥 ∈ 𝐵 ∩ 𝐶
Thus:
𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶)
Proven.
