MAT3702 Assignment 1 2026 (Answer Guide) – Due 13 May 2026

MAT3702 Assignment 1 2026 (Answer Guide) – Due 13 May 2026




ASSIGNMENT 1
Question 1
Prove that




Proof:

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
We prove the equality by showing mutual inclusion.




(⊆) Show that

A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)

Let x ∈ A ∪ (B ∩ C).

Then either:


1. x ∈ A, or
2. x ∈ B ∩ C.

Case 1: If x ∈ A, then clearly:
x ∈ A ∪ B, and
x ∈ A ∪ C.

, Thus x ∈ (A ∪ B) ∩ (A ∪ C).

Case 2: If x ∈ B ∩ C , then:

x ∈ B and
x ∈ C.
Therefore:

x ∈ A ∪ B, and
x ∈ A ∪ C.
Thus again x ∈ (A ∪ B) ∩ (A ∪ C).

Hence,

A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C)



(⊇) Show that

(A ∪ B) ∩ (A ∪ C) ⊆ A ∪ (B ∩ C)

Let x ∈ (A ∪ B) ∩ (A ∪ C).

Then:

x ∈ A ∪ B, and
x ∈ A ∪ C.
Thus:

x ∈ A or x ∈ B,
x ∈ A or x ∈ C.
If x ∈ A, then clearly x ∈ A ∪ (B ∩ C).

If x ∈
/ A, then:

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