Math 360, Fall 2017, 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.

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

1. Forward image (of a subset $S\subseteq A$ under a function $f:A\rightarrow B$).
2. Pre-image (of a subset $T\subseteq B$ under a function $f:A\rightarrow B$).
3. Fiber (of a function).
4. Inverse (of a relation).
5. Invertible (function).
6. Binary operation (on a set $S$).
7. Binary structure.
8. Ternary operation.
9. $n$-ary operation.
10. Nullary operation.
11. Associative (binary structure).
12. Commutative (binary structure).
13. Identity element (in a binary structure).
14. $\mathrm{Fun}(S,S)$.
15. $\mathbb{Z}_n$.

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

1. Criterion relating injectivity to fibers.
2. Criterion relating surjectivity to fibers.
3. Criterion relating bijectivity to fibers.
4. Theorem relating invertibility to bijectivity.
5. Theorem concerning uniqueness of identity elements.
6. Theorem concerning associativity of composition.

Solve the following problems:

1. Section 2, problems 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 17, 18, 19, 20, 21, 22, and 23.
2. Consider the binary operation $\triangle$ on $\mathbb{R}$ defined by the formula $a\triangle b = ab+a+b$. Does $(\mathbb{R},\triangle)$ have an identity element?
3. Repeat the problem above, this time for the operation $*$ defined by $a*b=ab+2a+2b+2$.

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