Math 360, Fall 2021, Assignment 4

Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different.

- Goethe

Read:

1. Section 4.

Carefully define the following terms, then give one example and one non-example of each:

1. $(\mathrm{Fun}(S,S))$.
2. $f\circ g$ (the composition of the functions $f$ and $g$).
3. $\iota$ (the identity function from a set $S$ to itself).
4. $\mathbb{Z}_n$.
5. $+_n$ (addition modulo $n$).
6. $\cdot_n$ (multiplication modulo $n$).
7. Semigroup.
8. Monoid.
9. Inverse (of an element of a monoid).
10. Group.
11. Abelian (group).

Carefully state the following theorems (you do not need to prove them):

1. Theorem regarding associativity of composition.
2. Theorem asserting that $+_n$ is well-defined.

Solve the following problems:

1. Section 4, problems 1, 3, 5, 11, 12, 13, 14, 18, and 19.
2. Give an example of (a) a binary structure which is not a semigroup, (b) a semigroup which is not a monoid, (c) a monoid which is not a group, and (d) a group which is not abelian.
3. By making an operation table, determine which elements of the commutative monoid $(\mathbb{Z}_5,\cdot_5)$ have inverses. Then do the same for $(\mathbb{Z}_6,\cdot_6)$ and $(\mathbb{Z}_8,\cdot_8)$.
4. Based on the results of the previous problem, try to make a conjecture regarding which elements of $(\mathbb{Z}_n,\cdot_n)$ have inverses.
5. Carefully show that the operation $\cdot_n$ is well-defined (i.e. state and prove a theorem analogous to the theorem asserting that $+_n$ is well-defined).

--------------------End of assignment--------------------

Theorems:

2. $+_n$ is well-defined: if $a=b mod(n)$ and $c = d mod(n)$, then $(a + c) = (b + d) mod(n)$.

Other Problem :

1. A binary structure which is not a semigroup: $(\mathbb Z, \triangle )$ , for $a, b \in \mathbb Z, a \triangle b = a + 2(b + 1)$;a semigroup which is not a monoid: $(S, \triangle), S = \{a,b\}$, for all $a, b \in S, a \triangle b = b$; a monoid which is not a group: $(Z_3, *)$, for $a, b \in Z_3, a * b = (a \cdot b) mod 3$; a group which is not abelian: $(S_5, *), S_5 = \{3*3 \text{ matrices whose determinant } \neq 0 \}$, $*$ is matrix multiplication.
2. Elements: $(\mathbb{Z}_5,\cdot_5): 1,2,3,4; (\mathbb{Z}_6,\cdot_6): 1,5; (\mathbb{Z}_8,\cdot_8): 1,3,5,7$.
3. for $n$, for all $a \in \mathbb Z$, if $a$ is not $1$ and $a$ divides $n$, $a$ and any $a * b for any b \in \mathbb Z$ will not have inverse
4. For operation $\cdot_n$