Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 41 — #49
2.4. Well Ordered Sets 41
Since we get a contradiction in both cases, we conclude that C must
be empty. That is, there are no counterexamples to (2.7), which proves
that (2.7) holds.
The proof makes an implicit assumption about the value of m. State the assumption
and justify it in one sentence.
Problem 2.13. (a) Prove using the Well Ordering Principle that, using 6¢, 14¢, and
21¢ stamps, it is possible to make any amount of postage over 50¢. To save time,
you may specify assume without proof that 50¢, 51¢, . . . 100¢ are all makeable, but
you should clearly indicate which of these assumptions your proof depends on.
(b) Show that 49¢ is not makeable.
Problem 2.14.
We’ll use the Well Ordering Principle to prove that for every positive integer n, the
sum of the first n odd numbers is n 2 , that is,
nX
1
.2i C 1/ D n 2 ; (2.8)
iD0
for all n > 0.
Assume to the contrary that equation (2.8) failed for some positive integer n. Let
m be the least such number.
(a) Why must there be such an m?
(b) Explain why m 2.
(c) Explain why part (b) implies that
mX
1
.2.i 1/ C 1/ D .m 1/ 2 : (2.9)
iD1
(d) What term should be added to the left-hand side of (2.9) so the result equals
mX
.2.i 1/ C 1/‹
iD1
(e) Conclude that equation (2.8) holds for all positive integers n.