Math 201-B
Introduction to Proofs
Instructor: Alex Roitershtein
Exam #2
November 13, 2015
Iowa State University
Department of Mathematics
Fall 2015
Student name:
- Duration of the exam 50 minutes.
- The exam includes 5 regular questions and a bonus problem.
- The total mark is 100 points for regular questions.
- Please show all the work, not only the answers.
- Calculators, textbooks, and help sheets are allowed.
Student ID:
1.
[20 points]
(a) Prove that if A, B, and C are sets such that
A ∩ ( B − C ) = ∅ and then
A ∩ ( C
( B ∪ C ) − ( B ∩ C ) ⊆ A.
− B ) = ∅ ,
(b) Prove that
Z
= { 2 m + 3 n : m, n ∈
Z
} .
2.
[20 points] Prove or disprove:
(a) A × B = C × D if and only if A = C and B = D.
(b) ( x, y ) ∈
R
2
: | x | + | y | ≤ 1 ⊆ ( x, y ) ∈
R
2
: x
2
+ y
2 ≤ 1 .
3.
[20 points] Using induction , show that ( n
3
+ 2 n ) is a multiplyer of 3 for all n ∈
N
.
4.
[20 points]
(a) Let R be a relation on
Z such that xRy if either x ≡ y (mod 3) or else x ≡ y (mod 5) .
Is R reflexive? Symmetric? Transitive?
(b) Let R be a relation on
Z such that xRy if 2 x + 5 y ≡ 0 (mod 3) .
Is R an equivalence relation?
5.
[20 points] Exercise 12 from Section 12.2 of the textbook BP.
6.
[Bonus] Is it possible to alternate just exactly one symbol/letter in the statement of Problem 4(b) in such a way that the answer will change to the opposite one?
1