Homework 2 (assigned June 29, due July 5)

1. Let m and n be integers. We say that m divides n, denoted m | n, if there exists an integer k such that n = km. Decide whether each of the following statements is true or false and justify your answer briefly. The universe for the variables m, n, and k is N, the set of positive integers. (2 points each)
   • (for all m) (for all n) (m | n)

   • (exists m) (for all n) (m | n)

   • (for all m) (exists n) (m | n)

   • (exists m) (exists n) (m | n)

   • (exists n) (for all m) (m | n)

   • (for all n) (exists m) (m | n)

2. Write down the converse of the following statement about integers:
   If x and y are odd, then x - y is even.

   Is the statement you wrote down true or false? Prove your answer. (4 points)

3. Exercises 8, 15, and 18 on page 48. (4 points each)

4. Exercises 2, 6, and 14 on page 60. (4 points each)

5. An arithmetic progression is a sequence of the form a, a + d, a + 2d, ..., a + nd, where a and d are real numbers. Use mathematical induction to prove that the sum of these terms of an arithmetic progression is given by
   a + (a + d) + (a + 2d) + ... + (a + nd) = (n + 1)(2a + nd)/2.

   (4 points)

6. Find two finite sets A and B such that A ∈ B and A ⊆ B. (3 points)

7. Can you conclude that A = B if A and B are two sets with the same power set? (3 points)

8. Exercise 61 on page 63. (5 points)

Gerry Brady
Wednesday June 29 20:23:19 CDT 2005