Problem 1. Use mathematical induction to prove that for all n ∈ ℕ,

0^{3} + 1^{3} + 2^{3} + ⋯ + (n − 1)^{3} < \frac{n^{4}}{4} and \frac{n^{4}}{4} < 1^{3} + 2^{3} + ⋯ + (n − 1)^{3} + n^{3}.

Use this result to compute the area of the region pictured below (the vertical distance between the point b units from 0 is b^{3}).

Problem 2. Let a, b ∈ ℝ with a ≠ 0. Use the field axioms of ℝ to carefully prove that if ax + b = 0 then x = \frac{−b}{a}. Justify all your steps.

Problem 3. Prove or disprove: For all propositions p, q, r, s, we have

((p ⇒ q) ⇒ r) ⇒ s ≡ ((p ∧ q) ∧ r) ⇒ s.

Problem 4. Let X and Y be sets. Recall, for sets X and Y, we define the set X \ Y to be

X \ Y = {x ∈ X satisfying x ∉ Y }.

Prove or disprove:

(a) For all sets A, B, C

(A \ B) ∪ C = (A ∪ C) \ (B ∪ C).

(b) For all sets A, B, C

A \ (B ∪ C) = (A \ B) ∪ (A \ C).

(c) For all sets A, B, C

A \ (B ∩ C) = (A \ B) ∪ (A \ C).

Problem 5. True or False: If S is a nonempty subset of rational numbers that is bounded above, then the least upper bound of S is rational.

Problem 6. Let X and Y be subsets of real numbers. Define the set X − Y to be

X − Y = {x − y where x ∈ X and y ∈ Y }.

(a) Suppose that X, Y ⊂ ℝ and that L is the least upper bound of X and M is the least upper bound of Y. Prove or disprove: the least upper bound of X − Y is L − M.

(b) Prove or disprove: For any sets X, Y ⊂ ℝ, if X ⊂ X − Y then 0 ∈ Y.

Problem 7. Use the order axioms of ℝ to prove that if x, y are positive real numbers with x < y then x^{2} < y^{2}.

Problem 8. More induction: use mathematical induction to prove that

(a) ∑_{k=1}^{n} = \frac{(n)(n + 1)}{2} for all n ∈ ℕ.

(b) n! > 2^{n} for all natural numbers n ≥ 4.

Problem 9. Prove that |x + y| ≤ |x| + |y| for all x, y ∈ ℝ.

Problem 10. Negate the following propositions:

(a) ∀x ∈ ℝ∃n ∈ ℕ(x < n)

(b) ∀x > 0∃n ∈ ℕ\frac{1}{n} < x

(c) ∀x ∈ ℝ∀y ∈ ℝ (x < y ⇒ ∃z ∈ ℝ(x < z < y))

EXAM

Practice Exam 1 Math 157 September 27, 2013

• You may use Mathematica (including the Mathematica Documentation) but you must start with an empty notebook.

• Absolutely no internet use is permitted during the exam.

• Each problem is worth 1 point for a total of 5 points, with a possible bonus point.

• Please mark your answers clearly on the answer key.

Success!