Chapter 1
Set Theory
Problem 1
(i) If any set has 2n elements, the number of its subsets will be 4^n.
(ii) Q = { p/q : p, q ∈ Z, q ≠ 0 } is the set of rational numbers.
(iii) If a, b ∈ R, then (a, b) = { x ∈ R : a < x < b }.
Which combination of these statements is correct?
Problem 2
Let Aₙ = { n, 2n, 3n, ... } for n ∈ N.
Which one is A₁ ∩ A₂?
Problem 3
Which one denotes A₃ ∩ A₆?
Problem 4
Which one equals A₂ ∩ A₃?
Problem 5
U = {1,2,...,20}
A = odd numbers
B = prime numbers
Which is correct?
Problem 6
The number of elements of sets A and B are shown in the Venn diagram. If n(A) = n(B), find (1) x (2) n(A ∪ B) (3) n(B \ A).
Problem 7
If U = {positive integers}, A = {x ≥ 5}, B = {5x < 12}, find n(A ∩ B) and n(A' ∪ B).
Problem 8
If U = even integers, A = {3x ≥ 25}, B = {5x < 12}, find n(A ∩ B).
Problem 9
Show that (1) A \ A = ∅ (2) A \ (A \ A) = A.
Problem 10
Show that A × (B ∪ C) = (A × B) ∪ (A × C).