1. Partitions

A partition of a set $S$ is a list of subsets $A _ 1,\dots, A _ n$ (for some $n$) so that

(i) $A _ 1 \cup A _ 2 \cup \dots \cup A _ n$ is all of $S$.

(ii) The $A _ i$ are pairwise disjoint, i.e., $A _ i \cap A _ j = \emptyset$ for $i \neq j$.

The subsets are called ``blocks'' or ``classes'' (an older term).

More generally, it is OK to have infinitely many blocks, in which case (i) can be expressed as $\bigcup _ i A _ i = S$.

Problem Q-2. In each of the following examples, say whether the subsets form a partition, and if they are not, say why not. (If they are a partition, no proof is needed.)

(a) $S$ is the set of seven plants in Problem G-9 and the blocks are the ones the problem asks for.

(b) $S$ is $\mathbb{R}^3$ and the blocks are a 2-dimensional subspace and all the planes parallel to it.

(c) $S$ is $\mathbb{R}^3$ and the blocks are a 1-dimensional subspace and all the lines parallel to it.

(d) $S$ is $\mathbb{R}^3$ and the blocks are all the 1-dimensional subspaces.

(e) $S$ is $\mathbb{Z}$ and there are two blocks, one consisting of all even integers and the other consisting of all odd integers.

(f) $S$ is $\mathbb{Z}$ and there are ten blocks, each consisting of integers that are congruent to each other modulo 10. For example, one of the blocks is $\{\dots, -13,-3,7,17,27,\dots \}$.

(g) $S$ is any set and the blocks are all the singleton subsets (1-element subsets).

(h) $f:S\rightarrow T$ is a function between sets, and the blocks are the inverse images of elements in the image [range] of $f$, in other words, the subsets $f^{-1} (t)$ for $t \in$   image$~f$.

(i) $S$ is Pols$(\mathbb{R})$ (polynomials of all degrees) and for each $r \in \mathbb{R}$ there is a block $A _ r = \{f \in$   Pols$(\mathbb{R}) \vert f(r) = 0\}$.

(j) $S = \mathbb{R}^2$ and for each $m,n \in \mathbb{Z}$ there is a block $A _ {m,n} = \{(x,y)\vert m \leq x < m+1, n \leq y < n+1\}$.

Problem Q-3. (a) Show that in a vector space $V$, the nonzero part $V \! \setminus \! \{$0$\}$ (meaning $V$ with $\{$0$\}$ omitted) is partitioned by the nonzero parts of the 1-dimensional subspaces of $V$.

(b) If $V$ is a 3-dimensional vector space over $GF(q)$, find a formula for the number of 1-dimensional subspaces of $V$.

(Method: Use (a). How large are the 1-dimensional subspaces? Your answer will be a polynomial in $q$. Does it check with the solution to Problem J-5? As mentioned before, $q$ will be a power of a prime, but that fact isn't needed for this problem.)