1 To be Graded
Problem 1. Let X = {1, 2, a}. Find the power set of X, P(X).
Problem 2. For each n ∈ N, let An = {(n + 1)k : k ∈ N}. (a) What is A1 ∩ A2? (b)
Determine the sets ∪{An : n ∈ N} and ∩{An : n ∈ N}.
Problem 3. Let A and B be two sets. Prove that A ⊆ B iff A ∩ B = A.
Problem 4. Let A, B and C be arbitrary sets. Prove that A∩(B ∪C) = (A∩B)∪(A∩C).
Problem 5. Let N be the set of natural numbers, and | be the relation of divisibility (i.e.
we say y ∈ N divides x ∈ N, denoted by y|x, if there exists an integer n such that x = ny).
Prove that | is an ordering relation on N.
Problem 6. Let ≤ be an ordering relation on the set X. We define the inverse of ≤, denoted
by ≥, as follows: ∀x, y ∈ X, x ≥ y if and only if y ≤ x. Prove that ≥ is an ordering relation
on X.
Problem 7. Let (X, ≤) be a totally-ordered space and Y ⊆ X an nonempty subset of X.
Let α be a lower bound of Y and β an upper bound of Y . Prove that α ≤ β.
2 Reading Assignments
• Review Lecture Notes # 1 and #2;
• Review Sections 1.1 and 1.2 of the textbook;
