Chapter 4

Question 1

Proof Template 11, Proof by contrapositive

Answer 1

To prove “If A, then B”: Assume (not B) and work to prove (not A).

Question 2

Proposition 20.1

Answer 2

Let R be an equivalence relation on a set A and let a, b ⋵ A. If a R/ (not a relation) b, then [a] ∩ [b] = ⦰

Question 3

Proposition 20.4

Answer 3

Let a and b be numbers with a ≠ 0. There is at most one number x with a*x + b = 0.

Question 4

Proposition 22.1

Answer 4

Let n be a positive integer. The sum of the first n odd natural numbers is n^2.

Question 5

Theorem 22.2 (Principle of Mathematical Induction)

Answer 5

Let A be a set of natural numbers. If 1. 0 is a member of A, and 2. ∀k is a member of set of Natural Numbers, k is a member of A then k +1 is a member of A, then A = set of Natural Numbers

Question 6

Proposition 22.7

Answer 6

Let n be a natural number. Then 4^n - 1 is divisible by 3.

Question 7

Theorem 22.9 (Principle of Mathematical Induction—strong version)

Answer 7

Let A be a set of Natural Numbers, if 1. 0 is a member of A and 2. for all k that is a member of set of Natural Numbers, if 0,1,2,……,k is a member of k, then k +1 is a member of A then A = the set of Natural Numbers

Question 8

Proposition 22.10

Answer 8

If a polygon with four or more sides is triangulated, then at least two of the triangles formed are exterior.