MAT3702
ASSIGNMENT 1
Abstract Algebra
Due date: 13 May 2026, Wednesday
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 5
, SOLUTIONS:
1. Let 𝐴, 𝐵, 𝐶 be sets. Show that 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
We prove equality by showing each side is contained in the other.
First, let 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶). Then 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 ∩ 𝐶.
If 𝑥 ∈ 𝐴, then 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶, so 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
If 𝑥 ∈ 𝐵 ∩ 𝐶, then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶. Thus 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶, so 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪
𝐶).
Hence 𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Conversely, let 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶). Then 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶.
If 𝑥 ∈ 𝐴, then clearly 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
Page 2 of 5
ASSIGNMENT 1
Abstract Algebra
Due date: 13 May 2026, Wednesday
FULL SOLUTIONS
COMPLETE SOLUTIONS
MEMORANDUM
UNISA 2026
Page 1 of 5
, SOLUTIONS:
1. Let 𝐴, 𝐵, 𝐶 be sets. Show that 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
We prove equality by showing each side is contained in the other.
First, let 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶). Then 𝑥 ∈ 𝐴 or 𝑥 ∈ 𝐵 ∩ 𝐶.
If 𝑥 ∈ 𝐴, then 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶, so 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
If 𝑥 ∈ 𝐵 ∩ 𝐶, then 𝑥 ∈ 𝐵 and 𝑥 ∈ 𝐶. Thus 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶, so 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪
𝐶).
Hence 𝐴 ∪ (𝐵 ∩ 𝐶) ⊆ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
Conversely, let 𝑥 ∈ (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶). Then 𝑥 ∈ 𝐴 ∪ 𝐵 and 𝑥 ∈ 𝐴 ∪ 𝐶.
If 𝑥 ∈ 𝐴, then clearly 𝑥 ∈ 𝐴 ∪ (𝐵 ∩ 𝐶).
Page 2 of 5
