Mathematical Reasoning- Writing and Proof, Version 2.1, 2014a

96 Chapter 3. Constructing and Writing Proofs in Mathematics
Exercises for Section 3.1
? 1. Prove each of the following statements:
(a) For all integers a, b, andc with a ¤ 0, ifa j b and a j c, then
a j .b c/.
(b) For each n 2 Z, ifn is an odd integer, then n 3 is an odd integer.
(c) For each integer a, if 4 divides .a 1/, then 4 divides a 2 1 .
2. For each of the following, use a counterexample to prove the statement is
false.
? (a) For each odd natural number n, ifn>3,then3divides n 2 1 .
(b) For each natural number n, .3 2 n C 2 3 n C 1/ is a prime number.
(c) For all real numbers x and y, p x 2 C y 2 >2xy.
? (d) For each integer a, if 4 divides a 2 1 ,then4divides.a 1/.
3. Determine if each of the following statements is true or false. If a statement
is true, then write a formal proof of that statement, and if it is false, then
provide a counterexample that shows it is false.
(a) For all integers a, b, andc with a ¤ 0, ifa j b,thena j .bc/.
? (b) For all integers a and b with a ¤ 0,if6 j .ab/, then6 j a or 6 j b.
(c) For all integers a, b, andc with a ¤ 0, ifa divides .b 1/ and a
divides .c 1/, thena divides .bc 1/.
? (d) For each integer n, if 7 divides n 2 4 ,then7divides.n 2/.
? (e) For every integer n, 4n 2 C 7n C 6 is an odd integer.
? (f) For every odd integer n, 4n 2 C 7n C 6 is an odd integer.
? (g) For all integers a, b, andd with d ¤ 0, ifd divides both a b and
a C b, thend divides a.
(h) For all integers a, b, andc with a ¤ 0, ifa j .bc/, thena j b or a j c.
? 4. (a) Ifx and y are integers and xy D 1, explain why x D 1 or x D 1.
(b) Is the following proposition true or false?
For all nonzero integers a and b, ifa j b and b j a, thena D˙b.
? 5. Prove the following proposition: