**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*), *f*^{2}(*a*), *f*^{3}(*a*), *f*^{4}(*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). ...

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