MAT3702
Assignment 1
Due 13 May 2026
, Question 1
Problem Statement
Let 𝐴, 𝐵, 𝐶 be sets. Show that
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Solution
Take an arbitrary element 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
This means either:
• 𝑥 ∈ 𝐴 , or
• 𝑥∈𝐵∩𝐶.
If 𝑥 ∈ 𝐴 , then clearly 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶 .
If 𝑥 ∈ 𝐵 ∩ 𝐶 , then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶 , so again
𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶 .
Hence,
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Now assume 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Then:
• 𝑥 ∈ 𝐴 ∪ 𝐵 , and
• 𝑥∈𝐴∪𝐶.
If 𝑥 ∈ 𝐴 , then clearly 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
If 𝑥 ∉ 𝐴 , then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶 , so
𝑥 ∈ 𝐵 ∩ 𝐶 , which again implies
𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
Assignment 1
Due 13 May 2026
, Question 1
Problem Statement
Let 𝐴, 𝐵, 𝐶 be sets. Show that
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Solution
Take an arbitrary element 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
This means either:
• 𝑥 ∈ 𝐴 , or
• 𝑥∈𝐵∩𝐶.
If 𝑥 ∈ 𝐴 , then clearly 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶 .
If 𝑥 ∈ 𝐵 ∩ 𝐶 , then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶 , so again
𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶 .
Hence,
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Now assume 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Then:
• 𝑥 ∈ 𝐴 ∪ 𝐵 , and
• 𝑥∈𝐴∪𝐶.
If 𝑥 ∈ 𝐴 , then clearly 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
If 𝑥 ∉ 𝐴 , then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶 , so
𝑥 ∈ 𝐵 ∩ 𝐶 , which again implies
𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
