MAT3707 Assignment 2 (710554 ) Detailed Solutions Due 18 July 2025

MAT3707
Assignment 2
Unique No: 710554
Due 18 July 2025

, MAT3707

Assignment 2: Detailed Solutions

Unique No: 710554

Due July 18, 2025


Question 1
Problem Statement. Given sets A with m elements and B with n elements:
[label=()]Compute |(A × {A}) ∪ (B × {B})| and compare with |A ∪ B|. Find |A×B|. Find
|2A|. Find |BA|. Find the number of k-element subsets of B; prove .

Step-by-step Solutions:
(i). Each element of A is paired with the singleton set {A} to form m ordered pairs in A×{A}.
Similarly, B ×{B} has n ordered pairs. These two sets are disjoint, so:

|(A × {A}) ∪ (B × {B})| = m + n.

For |A ∪ B|, if A and B have r elements in common, then:

|A ∪ B| = m + n − r.

Hence, |(A × {A}) ∪ (B × {B})| equals m + n, regardless of overlap.

(ii) Each element of A can be paired with every element in B. There are m choices from A
and n from B, so:

|A × B| = m · n.

(iii)The power set 2 A consists of all subsets of A. Each element has 2 choices (included or
not), so:
|2A| = 2m.
(iv) A function f : A → B assigns to each of the m elements in A one of the n values in B, so:
|BA| = nm.

(v) Number of k-element subsets of B is nk. Summing over k = 0 to n counts all subsets:

.