MAT3702 Assignment 1 Semester 1 2026 - Due 13 May 2026

MAT3702
ASSIGNMENT 1 2026
DUE: 13 MAY 2026 (MEMO)

, MAT 3702 Assignment 1 2026
Due Date: 13 May 2026

1. Let A, B, C be sets and show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).




Question 1


(⊆) Let x ∈ A ∪ (B ∩ C).
Then x ∈ A or x ∈ B ∩ C .

If x ∈ A, then x ∈ A ∪ B and x ∈ A ∪ C . Hence x ∈ (A ∪ B) ∩ (A ∪ C).
If x
∈ B ∩ C , then x ∈ B and x ∈ C . Hence x ∈ A ∪ B and x ∈ A ∪ C , so x ∈ (A ∪ B) ∩
(A ∪ C).

In both cases, x ∈ (A ∪ B) ∩ (A ∪ C). Thus A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C).

(⊇) Let 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 because x ∈ A ∪ B we must have x ∈ B . Similarly, from x ∈ A ∪ C we must
have x ∈ C . Therefore x ∈ B ∩ C , and hence x ∈ A ∪ (B ∩ C).

In both cases, x ∈ A ∪ (B ∩ C). Thus (A ∪ B) ∩ (A ∪ C) ⊆ A ∪ (B ∩ C).

each set contains the other, they are equal:



A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).