Discrete Structures, Logic, and Computability, 4th Edition

Chapter 9

Section 9.1

1. The zero is m because min(x, m) = min(m, x) = m for all x ∈ A. The identity is n because min(x, n) = min(n, x) = x for all x ∈ A. If x, y ∈ A and min(x, y) = n, then x and y are inverses of each other. Since n is the largest element of A, it follows that n is the only element with an inverse.

2. a. No; no; no. c. True; False; False is its own inverse.

3. S = {a, f (a), f²(a), f³(a), f⁴(a)}.

4. a. An element z is a zero if both row z and column z contain only the element z. c. If x is an identity, then an element y has a right and left inverse w if x occurs in row y column w and also in row w column y of the table.

5. a.

Notice that d ∘ b = a, but b ∘ d ≠ a. So b and d have one-sided inverses but not inverses (two-sided). ...