Math 201B
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