In Smith's Introductory Mathematics: Algebra and Analysis, I came across the definition of a transversal for a partition along with examples. Either I don't understand one of the examples, or it is simply wrong. Since I haven't authored any mathematics textbooks, I tend to think it's the former. Can someone clarify?
Here's the quote from the book:
Finally, when you have a partition of a set $S$, it is often useful to have one representative from each set comprising the partition. A set of these representatives is called a transversal for the partition. [...] We view a plane as a set, the elements of which are the geometric points comprising the plane. The plane can be partitioned into parallel straight lines, and then any straight line which is skew to the lines in the partition will suffice as a transversal. Such a skew straight line will intersect each of the family of parallel lines in exactly one point, as required. One could think of crazy transversals, where a point is selected from each of the family of parallel lines in some fashion, but our transversal has the virtue of being geometrically pleasant.
Here's my thoughts.
Let $L$ be the set of lines in the plane. There is an equivalence relation $\parallel \> : L \times L$ (does that functional notation properly describe relations, too?) on $L$. It gives rise to a partition of the set of lines into parallel lines, but I'm dubious that it partitions the plane because any point in the plane, say $(0,0)$, will be on infinitely many lines.
Wikipedia agrees that a transversal "is a set containing exactly one element from each member of the collection". It seems to me that if the collection in question is a partition of lines in the plane into (disjoint) sets of parallel lines, then an element of a member of the collection must be a line (i.e. an element of $L$ and not an element of $\mathbb{R}^2$). So if, as stated above, "a skew straight line will intersect each of the family of parallel lines in exactly one point" then I don't see how it is a transversal for the partition.
Based on my understanding, I propose that a transversal of the partition induced by $\parallel$ on lines in the plane is the set of lines in the plane through a given point, say $(0,0)$.
FYI, this is self-study not homework. Feel free to edit the tags if something is more appropriate.
First a minor notational point: no, the notation $\|:L\times L$ is not correct. Just say that the relation of being parallel is an equivalence relation on $L$. (Formally it’s a subset of $L\times L$ with certain properties.)
I suspect that you’re thinking of the partition of $L$ into equivalence classes of the relation $\|$; this is not what the author is talking about. He’s pointing out that each individual equivalence class of $\|$ is a partition of the plane into parallel lines.
If $\ell\in L$, let $[\ell]$ be the equivalence class of $\ell$, i.e., the set of all lines in the plane that are parallel to $\ell$. What the author is saying is that for each $\ell\in L$, the equivalence class $[\ell]$ is a partition of the plane. For example, for each $r\in\Bbb R$ let $\ell_r$ be the line whose equation is $x=r$, the set of all points $\langle r,y\rangle$ with first coordinate $r$. (E.g., $\ell_0$ is the $y$-axis.) Then $\{\ell_r:r\in\Bbb R\}$ is one of the equivalence classes: for each $t\in\Bbb R$, $[\ell_t]=\{\ell_r:r\in\Bbb R\}$. And as you can see, this family is a partition of the plane: if $\langle x,y\rangle\in\Bbb R^2$, $\ell_x$ is the unique member of the family containing $\langle x,y\rangle$.
What does a transversal for this partition look like? Since the members of $\{\ell_r:r\in\Bbb R\}$ are the lines parallel to the $y$-axis, a transversal for this partition is a subset $T$ of the plane that contains exactly one point on each line parallel to the $y$-axis. This means that for each $x\in\Bbb R$ there is exactly one $y_x\in\Bbb R$ such that $\langle x,y_x\rangle\in T$. This is actually a very familiar idea: this just says that $T$ is the graph of a function $f:\Bbb R\to\Bbb R$ given by $f(x)=y_x$. The graph of the function $f(x)=x^2$, for instance, is a transversal for this partition. So is the graph of $f(x)=2x$, or $f(x)=-3x+7$.
The last two graphs, of course, are straight lines that are skew to the parallel lines of the partition: the lines forming the partition are vertical, and the lines $y=2x$ and $y=-3x+7$ are not. Any non-vertical line in the plane — i.e., any line skew to the family of parallel vertical lines — has an equation of the form $y=ax+b$. For each value of $x$, that equation picks out a unique $y_x$, namely, the real number $ax+b$. In other words, for each $x$ the equation $y=ax+b$ picks out exactly one element of $\ell_x$, so its graph is a transversal for the partition $\{\ell_r:r\in\Bbb R\}$. This is what the author is getting at when he says that any straight line skew to the members of the partition is a transversal for it. But as we saw in the previous paragraph, the graph of every function $f:\Bbb R\to\Bbb R$ is a transversal for this particular partition, and most of these graphs are geometrically much less pleasant than a simple straight line.
In the foregoing discussion I worked with just one of the equivalence classes of the relation $\|$, the one consisting of the lines parallel to the $y$-axes, because it’s the easiest one to visualize and talk about. The other equivalence classes behave similarly, however.
Let me conclude by giving a complete enumeration of the equivalence classes of the relation $\|$. It’s convenient to change notation, however: for each $\theta\in[0,\pi)$ let $\ell_\theta$ be the line through the origin at an angle $\theta$ with the $x$-axis. Every point in the plane except the origin is on exactly one of these lines. Then $[\ell_\theta]$, the equivalence class of $\ell_\theta$, is the set of all lines in the plane parallel to $\ell_\theta$. If $\theta=\frac{\pi}4$, for instance, these are the lines with slope $1$, the lines whose equations are of the form $y=x+a$.
Every line in the plane is parallel to exactly one of the $\ell_\theta$ with $0\le\theta<\pi$, so $\{[\ell_\theta]:0\le\theta<\pi\}$ is a complete enumeration of the set of equivalence classes of $\|$.