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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