Math 360, Fall 2016, 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 $(S,\triangle)$).
3. Identity function.
4. Isomorphism (from one binary structure to another).
5. Isomorphic (binary structures).
6. Structural property.

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

1. Associativity of composition.
2. Uniqueness of identity elements.

Solve the following problems:

1. Section 3, problems 1, 2, 3, 4, 5, 6, 7, 8, and 9.
2. Suppose that $(S,\triangle)$ and $(T,*)$ are binary structures, and that $\phi$ is an isomorphism from $S$ to $T$. Prove that $\phi^{-1}$ is an isomorphism from $T$ to $S$. (Hint: the hardest part is showing that $\phi^{-1}(t_1*t_2) = \phi^{-1}(t_1)\triangle\phi^{-1}(t_2)$. But since $\phi$ is injective, it suffices to show that $\phi$ sends the two sides of this equation to the same element of $T$.)
3. Is every binary structure isomorphic to itself? If you think so, can you give a specific isomorphism?
4. If $S$ is isomorphic to $T$, will $T$ necessarily be isomorphic to $S$? Why or why not?
5. If $S$ is isomorphic to $T$ and $T$ is isomorphic to $U$, will $S$ be isomorphic to $U$? Why or why not?

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