Mathematics 242 – Principles of Analysis Makeup Midterm Exam 1 – October 7, 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

((∼ q) ∧ (p ⇒ q)) ⇒ ∼ p.

What does your result say?

II. Consider the following definition: A function f is strictly increasing if and only if for all real x and all real y, if x < y then f(x) < f(y).

A) (5) Consider f : R → R defined by f(x) = x^{2}. What is wrong with the following

reasoning: if x < y, then x^{2} < y^{2}; therefore f(x) = x^{2} is a strictly increasing function. B) (10) Give the negation of the statement “f is strictly increasing”.

III. Let A = {x ∈ R : 0 < x^{2} − 1 < 3}, let B be the interval [−1, 2), and let C = {−1, 2}.

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) = 2x + 1. 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 set of positive and negative integers, Z, is denumerable.

V. (15) Prove that for all n ∈ N,

1 + 3 + 5 + ⋯ + (2n − 1) = n^{2}.

Extra Credit

(10) Show that if f : A → B and g : B → C are injective mappings, then g ∘ f : A → C is also injective.