Exercises for self study. Complete these exercises in the textbook and
compare your solutions against those in the back of the book. These exercises
will not be graded.
pp. 350–352, Exercises 3, 23 and 41
p. 362, Exercise 3
Exercises to be graded. Complete these exercises and turn in your
solutions.
It is easy to prove that the product of two consecutive integers is
even by a proof by cases. The goal of this exercise is to prove
this fact by induction. Let $p(n)$ denote the predicate that
$n(n + 1)$ is even for an integer variable $n$.
(a) What is the smallest nonnegative integer $n_0$ such that $p(n_0)$ is
true? Prove your claim.
(b) Give an inductive hypothesis, in a complete statement, of a proof by
induction that $p(n)$ is true for all integers $n \ge n_0$.
(c) Given such an inductive hypothesis, what do you need to prove in the
inductive step of the proof? Give a complete statememt.
(d) Complete the inductive step of the proof, clearly identifying where the
inductive hypothesis is used.
(e) Conclude, with justification, the proof that $p(n)$ is true for all
integers $n \ge n_0$.
Let $p(n)$ denote the predicate that
$$
\left(1 + \frac{1}{1} \right) \left(1 + \frac{1}{2} \right) \cdots
\left(1 + \frac{1}{n} \right) = n + 1
$$
for an integer variable $n$. First, determine the smallest positive
integer $n_0$ such that $p(n_0)$ is true, and then prove by induction that
$p(n)$ is true for all integers $n \ge n_0$.
Let $p(n)$ denote the predicate that a postage of $n$¢ can
be formed using only 3¢ and 10¢ stamps for
an integer variable $n$. First, determine the smallest integer amount $n_0$
such that $p(n)$ is true for $n = n_0, n_0 + 1, n_0 + 2$, and then prove by
strong induction that $p(n)$ is true for all integer amounts $n \ge n_0$.
Plagiarism and academic dishonesty.
Remember, under any circumstance, you must not copy part or all of another's
work and present it as your own. For more details, read
our course policy on plagiarism and
academic dishonesty.
