Math 360, Fall 2015, Assignment 3

No doubt many people feel that the inclusion of mathematics among the arts is unwarranted. The strongest objection is that mathematics has no emotional import. Of course this argument discounts the feelings of dislike and revulsion that mathematics induces....

- Morris Kline, Mathematics in Western Culture

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

1. Binary structure.
2. Identity element (in a binary structure $(A,*)$).
3. $\mathrm{Fun}(S,S)$.
4. Identity function.
5. $\mathbb{Z}_n$.
6. Addition modulo $n$.
7. Isomorphism (from one binary structure to another).

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

1. Uniqueness of identity elements.
2. Associativity of composition.
3. Theorem showing that addition modulo $n$ is a well-defined operation (also, please be prepared to give an example of an "operation" that looks well-defined but isn't).
4. Associativity of modular addition.

Solve the following problems:

1. Section 3, problems 1, 3, 7, 8, and 9.
2. Prove that the identity function $\iota:S\rightarrow S$ is an identity element for the binary structure $(\mathrm{Fun}(S,S),\circ)$.
3. Define a binary operation $\cdot_n$ on $\mathbb{Z}_n$, called multiplication modulo $n$, by the formula $$[a]_{\equiv_n} \cdot_n [b]_{\equiv_n} = [ab]_{\equiv_n}.$$ Prove that $\cdot_n$ is a well-defined operation. (Hint: mimic the proof that $+_n$ is well-defined, and do not panic when this proof turns out to be slightly more complicated. In particular, instead of simply multiplying the two equations, first solve them for $a_2$ and $b_2$ respectively, then multiply, then rearrange as necessary.)
4. Write the operation table for the structure $(\mathbb{Z}_4,\cdot_4)$.
5. Prove that $(\mathbb{Z}_n,\cdot_n)$ is associative and commutative, and has an identity element.

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