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Math 201-B Introduction to Proofs Instructor: Alex Roitershtein Iowa State University Department of Mathematics Fall 2015 Exam #1 October 7, 2015 Student name: - Student ID: Duration of the exam 50 minutes. The exam includes 5 questions. The total mark is 100 points. Please show all the work, not only the answers. Calculators, textbooks, and help sheets are allowed. 1. [20 points] (a) Find (A ∩ B) × (A − B) if A = {a, b}, B = {b, c}. (b) Negate the following statement: “Every integrable on [0, 1] function is continuous on the open interval (0, 1)”. 2. [20 points] Let A, B, C be arbitrary sets. Verify the following set identities using Venn diagrams: (a) (A ∪ B) ∩ C = (A − C) ∪ (B − C). (b) (A ∪ B) − (A ∩ B) = (A − B) ∪ (B − A). 3. [20 points] (a) Write a truth table for the logical statement (∼ P ∨ Q) ∧ (∼ Q ∨ R) ∧ (∼ R ∨ P ). (b) Decide whether or not the following two statements are logically equivalent: ∼ (∼ P ∨ Q) ∧ (∼ Q ∨ R) and ∼ P ∨ ∼ R. 4. [20 points] (a) How many three-digits positive integers 100a + 10b + c (here a, b, c are the digits) have the following property: . (a − b + c) .. 11 (a − b + c divides 11) 1 ? (b) How many three-digits positive integers either have the above property or are multiples of 10? 5. [20 points] It is easy to verify that 32 + 42 + 122 = 132 . Correspondingly, we say that (x, y, z, w) = (3, 4, 12, 13) is a solution in integers of the equation x2 + y 2 + z 2 = w 2 . (1) (a) Is it true that (1) has infinitely many integer solutions? (b) Is there any integer solution (x, y, z, w) to (1) such that x = y = 1? (c) Show that if (x, y, z, w) is a solution to (1) then at least one of the four numbers x, y, z, w divides 3. Hint: show first that for any x ∈ Z, either x2 = 3k or x2 = 3k + 1 for some k ∈ Z. 2