MAT3707 Assignment 2 (710554 ) Due 18 July 2025

MAT3707
Assignment 2
Unique code: 710554
Due 18 July 2025

,MAT3707 – Assignment 2
Unique No.: 710554

Due Date: 18 July 2025

Question 1
Let A be a set with m elements and B a set with n elements.

Part (i): Number of Elements in (A × {A}) ∪(B × {B}) and Comparison with A ∪B

Step 1: Understanding the Sets
Let:
- A = { a₁ , a₂ , ..., aₘ } so |A| = m
- B = { b₁ , b₂ , ..., bₙ } so |B| = n
Now consider the Cartesian products:
- A × {A} = { (aᵢ, A) : aᵢ ∈ A }
- B × {B} = { (bⱼ, B) : bⱼ ∈ B }

Step 2: Compute |A × {A}|
Since |A| = m and |{A}| = 1 (a singleton set), we calculate:
|A × {A}| = m × 1 = m

Step 3: Compute |B × {B}|
Similarly, since |B| = n and |{B}| = 1:
|B × {B}| = n × 1 = n

Step 4: Compute the Union
Since A × {A} and B × {B} are disjoint (distinct first elements and distinct second
elements A ≠ B),
we can compute:
|(A × {A}) ∪ (B × {B})| = m + n

Step 5: Compare with A ∪B
A ∪ B includes only distinct elements from A and B:
|A ∪ B| = m + n – k, where k = |A ∩ B|
So |A ∪ B| ≤ m + n with equality only if A ∩ B =∅

, Step 6: Example
Let A = {1, 2, 3} and B = {3, 4}
Then:
- A × {A} = {(1, A), (2, A), (3, A)} → size = 3
- B × {B} = {(3, B), (4, B)} → size = 2
Union size: 3 + 2 = 5
A ∪ B = {1, 2, 3, 4} → size = 4

Conclusion
(A × {A}) ∪ (B × {B}) contains ordered pairs and has size m + n.
A ∪ B contains plain elements and ma y be smaller if there’s overlap.
These are fundamentally different sets.