Math 360, Fall 2020, 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

Read:

1. Section 2.
2. Section 3.

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

1. Binary operation (on a set $S$).
2. Binary structure.
3. Commutative (binary structure).
4. Associative (binary structure).
5. Identity element (in a binary structure).
6. $(\mathrm{Fun}(S,S))$.
7. $f\circ g$ (the composition of the functions $f$ and $g$).
8. $\iota$ (the identity function from a set $S$ to itself).
9. Isomorphism (from a binary structure $(S,\triangle)$ to another binary structure $(T,*)$).

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

1. Theorem concerning the associativity of composition.

Solve the following problems:

1. Section 2, problems 1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 13, 17, 18, 19, 20, 21, 22, and 23.
2. Section 3, problems 2, 3, 4, 6, 7, and 17.
3. (Inverse elements) Suppose $(S,\triangle)$ is a binary structure with two-sided identity $e$, and let $s\in S$ be an arbitrary element. An inverse for $s$ is another element $s'\in S$ such that $s\triangle s' = e$ and $s'\triangle s = e$. Working in the binary structure $(\mathbb{Z},+)$, compute the inverse of $1$, the inverse of $-2$, and the inverse of $3$. In general, which elements have inverses, and can you make a formula that describes the inversion process?
4. Repeat the above problem for the binary structure $(\mathbb{Z},\cdot)$.
5. (Units) Let $(S,\triangle)$ and $e$ be as above. An element with an inverse is called a unit of $(S,\triangle)$. Describe the units of $(\mathbb{Z},\cdot)$, of $(M_n(\mathbb{R}),\cdot)$, and of $(\mathrm{Fun}(S,S),\circ)$ (where $S=\{a,b\}$ is a two-element set).
6. (Challenge) Can you describe the units of $(\mathrm{Fun}(S,S))$ when $S$ is an arbitrary set?

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

Questions:

Question 3 supposes there is only one two-sided element per bianry structure. Is this true? How might one prove it?

Excellent question! Suppose we have two two-sided identity elements in the same binary structure $(S,\triangle)$. Call them $e$ and $e'$. Now consider the element $e\triangle e'$. Since $e$ is an identity element, we must have $e\triangle e'=e'$. On the other hand, since $e'$ is an identity element, we must have $e\triangle e'=e$. Thus $e=e'$. (In general, to show that there is at most one of something, suppose you have two of that thing, and show that the two must be equal.) -Steven.Jackson (talk) 15:22, 29 September 2020 (EDT)

Solutions:

Binary operation (on a set SS). Let S be any set a binary operation on S id a function. f: SxS \mapsto S Binary structure.

Commutative (binary structure).

Associative (binary structure).

Identity element (in a binary structure). (Fun(S,S)) (Fun(S,S)).

f∘g

f∘g

(the composition of the functions ff and gg).

ιι(the identity function from a set SS to itself). Isomorphism (from a binary structure (S,△)(S,△) to another binary structure (T,∗)(T,∗)).