Reading Quiz Section 5.2
1. Which of the following statements are true? Select all that apply.
(a) Every well-ordered set of real numbers has a minimum element.
(b) If a set of real numbers has a minimum element, then it is well-ordered.
(c) Any finite set of real numbers is well-ordered.
(d) Induction proofs must have a base case of 0 or 1.
2. True or false: the set A = {
1
n
: n ∈ N} is well-ordered.
3. For each n ∈ N, let P(n) and Q(n) be propositions.
(a) Let s be the smallest natural number such that P(s) is false. What can you say about the
elements of the set A = {n ∈ N : n < s} with respect to the property P?
(b) Let a be minimal such that P(a) ∨ Q(a) is false. What can you say about the elements of
the set B = {n ∈ N : n < a} with respect to the properties P and Q?
(c) Let u be minimal such that P(u) ∧ Q(u) is false. What can you say about the elements of
the set C = {n ∈ N : n < u} with respect to the properties P and Q?
(d) Assume that P(1) is true, but that “∀n ∈ N, P(n)” is false. Explain why there exists a
natural number k such that the implication P(k) =⇒ P(k + 1) is false.
4. Here is an argument attempting to justify
n
∑
i =1
i =
1
2
n(n + 1) + 7. What is wrong with it?
Assume that the statement is true for some fixed n. Then
n+1
∑
i =1
i =
n
∑
i =1
i + (n + 1) =
1
2
n(n + 1) + 7 + (n + 1) =
1
2
( n + 1)[(n + 1) + 1] + 7
hence the statement is true for n + 1 and, by induction, for all n ∈ N.
Practice Problems Section 5.1
1. Prove that n! > 2
n
for all n ≥ 4.
Video Solution
