### Never Give Up. Never Surrender.

4. Let $X = \{0,1,2,3\}$ again; compute $\wp(X)$ (the power set of $X$).

OK. First of all, that’s a “Weierstrass pay” not a script-P; typesetting is always harder than it looks. There’s a “caligraphy” font available I think. But why should I have to know or care? One curly-P looks an awful lot like another and for all I can tell nobody is paying any attention literally.

Anyhow. In context, the power set of {0, 1, 2, 3} is
{
{},
{0}, (1}, {2}, (3},
{0,1}, {0,2}, {0.3},
{1,2}, {1,3}, {2,3},
{1,2,3}, {0,2,3}, {0,1,2}, {1,2,3},
{1,2,3,4}
}
.

Nobody in my class took advantage of my type-oriented “hints”… the carriage-return style I’ve used here (as I also typically do in blackboard work)… and no damn wonder. It’d be helpful if textbooks would talk about how things are actually done in practice but then maybe it would be helpful if everybody clapped for tinkerbell while we’re at it.

The main thing to see here is that, for example, every set-of-three (in this context) is the “complement” of a set-of-one… and so, for example, the set {1,2,3}… a “three-set”… must be in the power set precisely because the set {0} (a “one-set”) has already been “counted”. The same “principle of complementation” can be used more generally to show that, in a Universal Set of n elements, one has exactly as many k-element subsets as there are subsets with n-k elements.

Another good trick in this context is “binary notation”. Many of my students were well aware that the initial string of the natural numbers can usefully be written as (( 0000, 0001, 0010, … 1111))… standard “binary” notations for the natural number sequence ((0, 1, 2, … 15)). By a more-or-less-obvious “place value” notation (order the set; “0” means leave-it-out and “1” means put-it-in), we can set up a one-to-one correspondence of subets-of-{0,1,2,3) [or any other 4-element-set] and the binary “strings” 0000 … 1111). It pleases me immensely that (a handful of) students checked their work on this exercise by this technique explicitly.

5. Let $Y = \{0,3,6\}$ and $Z = \{0,5\}$ again;
compute {\bf a.} $Y\times Z$ and {\bf b.} $Z \times Y$.

6. Let $U = \{0, 1, 2, 3, 4, 5, 6\}$ again.
Write out the Relation $R$ defined on $U$ by $xRy \equiv 3 | (x - y)$.

Let $f(x) = log_3(x)$, $g(x) = 5x+1$. Find (“formulas” for) the following functions.
$f^{-1}$
$g^{-1}$
$f\circ g$
$g \circ f$