Math 360, Fall 2014, Assignment 2

We admit, in geometry, not only infinite magnitudes, that is to say, magnitudes greater than any assignable magnitude, but infinite magnitudes infinitely greater, the one than the other. This astonishes our dimension of brains, which is only about six inches long, five broad, and six in depth, in the largest heads.

- Voltaire

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

1. Binary operation (on a set $S$).
2. Induced operation (on a subset of $S$).
3. Commutative operation.
4. Associative operation.
5. Composition (of two functions).

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

1. Theorem relating equivalence relations to partitions (Theorem 0.22).
2. Associativity of composition (Theorem 2.13).

Solve the following problems:

1. Section 0, problems 29, 31, and 34.
2. Section 2, problems 2, 3, 7, 8, 9, 10, and 11.

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

Solutions:

Definitions:

1. A binary operation on a set $S$ is a function from $S\times S$ into $S$.
   1. Example: Normal addition on the reals is a binary operation. We write $+(a,b)$ as $a+b$.
   2. Nonexample: Subtraction on $\mathbb{Z}^+$ is not a binary operation. $-$ maps $(3,4)$ to $-1\not\in\mathbb{Z}^+$.
2. Suppose $\ast$ is a binary operation on a set $S$ and $A\subseteq S$. Then $\ast$ induces an operation on $A$ if and only if for all $a_1,a_2\in A$, $(a_1\ast a_2)\in A$. In this case the operation on $A$ obtained by restricting the domain of $*$ is said to be induced by $*$ (and is usually denoted by the same symbol $*$).
   1. Example: Ordinary addition on $\mathbb{Z}$ is induced by ordinary addition on $\mathbb{R}$.
   2. Non-example: Addition modulo $n$ on $\{0,1,2,\dots,n-1\}$ is not induced by ordinary addition on $\mathbb{Z}$.