MAT3702
Assignment 1
Due 13 May 2026
, Question 1
Problem Statement
Let𝐴, 𝐵, 𝐶 be sets. Show that
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Approach
To prove two sets are equal, we show that each set is contained in the other.
First Inclusion
Assume𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
This means either
• 𝑥 ∈ 𝐴 , or
•𝑥 ∈ 𝐵 ∩ 𝐶
If 𝑥 ∈ 𝐴 , then clearly
𝑥 ∈ 𝐴 ∪ 𝐵 and𝑥 ∈ 𝐴 ∪ 𝐶 .
If 𝑥 ∈ 𝐵 ∩ 𝐶 , then𝑥 ∈ 𝐵 and𝑥 ∈ 𝐶 , so again
𝑥 ∈ 𝐴 ∪ 𝐵 and𝑥 ∈ 𝐴 ∪ 𝐶 .
Hence
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
So
𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Second Inclusion
Now assume
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Assignment 1
Due 13 May 2026
, Question 1
Problem Statement
Let𝐴, 𝐵, 𝐶 be sets. Show that
𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Approach
To prove two sets are equal, we show that each set is contained in the other.
First Inclusion
Assume𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
This means either
• 𝑥 ∈ 𝐴 , or
•𝑥 ∈ 𝐵 ∩ 𝐶
If 𝑥 ∈ 𝐴 , then clearly
𝑥 ∈ 𝐴 ∪ 𝐵 and𝑥 ∈ 𝐴 ∪ 𝐶 .
If 𝑥 ∈ 𝐵 ∩ 𝐶 , then𝑥 ∈ 𝐵 and𝑥 ∈ 𝐶 , so again
𝑥 ∈ 𝐴 ∪ 𝐵 and𝑥 ∈ 𝐴 ∪ 𝐶 .
Hence
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
So
𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
Second Inclusion
Now assume
𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶)
