Mathematics 242 – Principles of Analysis Midterm Exam 1 – October 1, 2004

Directions: Do all work in the blue exam booklet. There are 100 possible regular points (distributed as indicated in the questions below), and 10 Extra Credit points.

I. (10) Construct the truth table for the compound expression

(p ⇒ q) ⇔ (∼ p ∨ q).

What does your result say?

II. Consider the following definition: A function f is periodic if and only if there exists a real number P > 0 such that for all real numbers x, f(x + P) = f(x).

A) (5) Consider f : R → R defined by f(x) = x^{2}. If P = 2, then notice f(−1+2) = f(−1)

is true. Does this say f(x) = x^{2} is periodic? Why or why not? B) (10) Give the negation of the statement “f is periodic”.

III. Let A = {x ∈ R : 1 < x^{2} < 9}, let B be the interval [−2, 4), and let C = {1, 2, 3}.

A) (10) What is the set A ∩ B? B) (10) What is the set (A ∪ B) \ C? C) (10) Let f : R → R be defined by f(x) = −x + 3. What is the set f^{−1}(A)? D) (10) Is the function f from part C injective? Is it surjective (onto R)? Justify your

answers.

IV. (20) Show that the open interval (0, 1) in the real numbers is not a denumerable set.

V. (15) Prove that for all n ∈ N,

1^{3} + 2^{3} + ⋯ + n^{3} = \frac{1}{4}n^{2}(n + 1)^{2}.

Extra Credit

(10) Let x be a non-negative real number. Show that if x ≤ ε for all ε > 0, then x = 0.