epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

44 The Nuts and Bolts of Proof, Third Edition
-nh(xi)> -nh(x2). Thus, /c„(xi)>/c„(x2), which proves that /c„ is a
decreasing function.
Part 4. If 4, then 1.
By definition of the functions used,/(x) = fei(x). Thus, this impHcation is
trivially true.
The proof is now complete. •
EXERCISES
Prove the following statements.
1. A function/is nonincreasing if and only if ^^^lz{^^^ S 0 for all c and
X in the domain of / with x^ c. (See front material of the book
for the definition of nonincreasing function.)
2. The product of two integers is odd if and only if they are both odd.
3. Let n be a positive integer. Then n is divisible by 3 if and only if n^ is
divisible by 3.
4. Let r and s be two counting numbers. The following statements are
equivalent:
i. r>s.
ii. a^<af for all real numbers a>l.
iii. d <a^ for all real positive numbers a< 1.
5. Let a and h be two distinct real numbers. The following statements are
equivalent:
i. The number h is larger than the number a.
ii. Their average, {a + b)/2, is larger than a.
iii. Their average, {a + ft)/2, is smaller than h.
Prove this statement by proving "If i, then ii," "If ii, then iii," and
"If iii, then i."
6. Let X and y be two distinct negative real numbers. The following
statements are equivalent:
i. x<y.
ii. \x\>\y\.
iii. x^>y^.
7. Consider the two systems of linear equations:
a\x-\- h\y = c\
' aix + h^y = C2