Set Theory Exercise 1.1

Problem 1: Cartesian Products
Find A×B, A×A and B×A for:
(i) A = {−2, 2, 3} and B = {−1, 4}
(ii) A = B = {p, q}
(iii) A = {m, n}; B = ∅
(i)
A×B = {(-2,-1), (-2,4), (2,-1), (2,4), (3,-1), (3,4)}
A×A = {(-2,-2), (-2,2), (-2,3), (2,-2), (2,2), (2,3), (3,-2), (3,2), (3,3)}
B×A = {(-1,-2), (-1,2), (-1,3), (4,-2), (4,2), (4,3)}
(ii)
Since A = B, all products are identical:
A×B = A×A = B×A = {(p,p), (p,q), (q,p), (q,q)}
(iii)
A×B = ∅ (since B is the empty set)
A×A = {(m,m), (m,n), (n,m), (n,n)}
B×A = ∅
Key Concept: The Cartesian product with the empty set is always the empty set.
Problem 2: Prime Numbers
Let A = {1,2,3} and B = {x | x is a prime number less than 10}.
Find A×B and B×A.
First, identify set B:
Prime numbers less than 10: {2, 3, 5, 7}
Now compute A×B:
A×B = { (1,2), (1,3), (1,5), (1,7), (2,2), (2,3), (2,5), (2,7), (3,2), (3,3), (3,5), (3,7) }
Now compute B×A:
B×A = { (2,1), (2,2), (2,3), (3,1), (3,2), (3,3), (5,1), (5,2), (5,3), (7,1), (7,2), (7,3) }
Key Observation: A×B ≠ B×A unless A = B. The order matters in Cartesian products.
Problem 3: Find Sets from Product
Given B×A = {(-2,-3), (-2,4), (0,-3), (0,4), (3,-3), (3,4)},
determine the sets A and B.
In the Cartesian product B×A:
- First elements come from set B
- Second elements come from set A
From the given pairs:
Distinct first elements: {-2, 0, 3} ⇒ B = {-2, 0, 3}
Distinct second elements: {-3, 4} ⇒ A = {-3, 4}
Solution: A = {-3, 4}, B = {-2, 0, 3}
Problem 4: Cartesian Equality
If A = {5,6}, B = {4,5,6}, C = {5,6,7},
Show that A×A = (B×B) ∩ (C×C).
First find A×A:
A×A = {(5,5), (5,6), (6,5), (6,6)}
Find B×B:
B×B = { (4,4), (4,5), (4,6), (5,4), (5,5), (5,6), (6,4), (6,5), (6,6) }
Find C×C:
C×C = { (5,5), (5,6), (5,7), (6,5), (6,6), (6,7), (7,5), (7,6), (7,7) }
Find (B×B) ∩ (C×C):
Common pairs: {(5,5), (5,6), (6,5), (6,6)}
This exactly matches A×A, proving the equality.
Problem 5: Set Equality
Given A = {1,2,3}, B = {2,3,5},
C = {3,4} and D = {1,3,5},
Verify if (A∩C) × (B∩D) = (A×B) ∩ (C×D)
First find A∩C:
A∩C = {1,2,3} ∩ {3,4} = {3}
Find B∩D:
B∩D = {2,3,5} ∩ {1,3,5} = {3,5}
Compute (A∩C) × (B∩D):
= {3} × {3,5} = {(3,3), (3,5)}
Now compute (A×B) ∩ (C×D):
A×B has 9 elements, C×D has 6 elements
Intersection: {(3,3), (3,5)}
Both sides equal {(3,3), (3,5)}, so the equality holds.
Problem 6: Distributive Properties
Let A = {x ∈ W | x < 2},
B = {x ∈ ℕ | 1 < x ≤ 4},
C = {3,5}.
Verify that:
(i) A × (B∪C) = (A×B) ∪ (A×C)
(ii) A × (B∩C) = (A×B) ∩ (A×C)
(iii) (A∪B) × C = (A×C) ∪ (B×C)
First identify the sets:
A = Whole numbers less than 2 = {0,1}
B = Natural numbers 1 < x ≤ 4 = {2,3,4}
C = {3,5}
(i) Verification
B∪C = {2,3,4,5}
A × (B∪C) = { (0,2), (0,3), (0,4), (0,5), (1,2), (1,3), (1,4), (1,5) }
(A×B) ∪ (A×C) produces the same set
(ii) Verification
B∩C = {3}
A × (B∩C) = {(0,3), (1,3)}
(A×B) ∩ (A×C) = {(0,3), (1,3)}
(iii) Verification
A∪B = {0,1,2,3,4}
(A∪B) × C = 10 elements
(A×C) ∪ (B×C) produces identical set
Key Concept: Cartesian products distribute over union and intersection.
Problem 7: Natural Numbers
Let A = The set of all natural numbers less than 8,
B = The set of all prime numbers less than 8,
C = The set of even prime number.
Verify that:
(i) (A∩B) × C = (A×C) ∩ (B×C)
(ii) A × (B−C) = (A×B) − (A×C)
First identify the sets:
A = {1,2,3,4,5,6,7}
B = {2,3,5,7}
C = {2} (only even prime)
(i) Verification
A∩B = B = {2,3,5,7}
(A∩B) × C = {(2,2), (3,2), (5,2), (7,2)}
(A×C) ∩ (B×C) produces identical set
(ii) Verification
B−C = {3,5,7}
A × (B−C) has 21 elements (7×3)
(A×B) − (A×C) produces identical set
Both properties are verified to hold true.