Math 201-B Iowa State University Introduction to Proofs Department of Mathematics

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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
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