1. Define $V_0=\emptyset$, $V_1={\mathcal P}(V_0)$, $V_2={\mathcal P}(V_1)$, and $V_3={\mathcal P}(V_2)$.
(a) Express the sets $V_0, V_1, V_2$ and $V_3$ in roster notation.
(b) Draw a directed graph whose ``dots'' (or ``vertices'')) are the sets in $V_3$ and whose directed edges, written $x\to y$, are interpreted to mean $x\in y$. (Hint: your graph should have four ``dots'' and four edges.)

2. Find sets $A$ and $B$ satisfying the given conditions.
(a) $A\in B$ and $A\not\subseteq B$.
(b) $A\in B$ and $A\subseteq B$.
(c) $A\notin B$ and $A\subseteq B$.

3. Show that $\bigcup {\mathcal P}(x) = x$. (Remember the proper way to show two sets are equal!)

Solution sketches (dischw1sol.tex),
Solution sketches (dischw1sol.pdf),

1. We have explained why the Russell class ${\mathcal R}=\{x\;|\;x\notin x\}$ is not a set. Explain why the following classes are also not sets.
(a) The class $\mathcal C$ of all sets.
(b) The class $\mathcal D$ of all $1$-element sets.
(Hints: For (a), show that the assumption that $\mathcal C$ is a set allows you to construct $\mathcal R$ as a set. For (b), show that the assumption that $\mathcal D$ is a set allows you to construct $\mathcal C$ as a set.)

2. Your friend offers a wager that, under the Kuratowski encoding, the ordered pair $(0,1)$ equals the natural number three. Should you take the wager? Explain.
(More detail: You friend is offering money to support the claim that the set $(0,1)$ equals the set $3$. If you believe that your friend is correct, then you should not take the wager. Otherwise, to take the wager, you should offer money saying that this is false. Whoever is correct will get all the money.)

3. Show that $\emptyset\times A = \emptyset$.

Solution sketches (dischw2sol.pdf),

Make sure you know all the information on the Quizlet card sets called Axioms of Set Theory, Vocabulary for Set Theory, Set Theory Examples, and Vocabulary for Functions.

1. Explain why it is true that the function $F: A\to {\mathcal P}(A): a\mapsto \{a\}$ is injective.

2. In this problem, $f: A\to B$ and $g:B\to C$ will be composable functions.
(a) Show that if $g\circ f$ is injective, then $f$ is injective.
(b) Show that if $g\circ f$ is surjective, then $g$ is surjective.

3. This is a continuation of Problem 2, so assume that $f: A\to B$ and $g:B\to C$ are composable functions.
(a) Give an example where $g\circ f$ is injective, but $g$ is not injective.
(b) Give an example where $g\circ f$ is surjective but $f$ is not surjective.

Solution sketches (dischw3sol.pdf).

Make sure you know all the information on the Quizlet card sets called Axioms of Set Theory, Vocabulary for Set Theory, Set Theory Examples, and Vocabulary for Functions.

1. Exercise 2.5.4(7) parts (a) and (b). (Here is a screenshot of Exercise 2.5.4(7).)

2. For each of (a)-(c), give an example of a binary relation $R$ on some set $A$ where
(a) $R$ is reflexive and symmetric, but not transitive.
(b) $R$ is reflexive and transitive, but not symmetric.
(c) $R$ is symmetric and transitive, but not reflexive.

3. Explain why the following statement is true: A function is injective if and only if its kernel is the equality relation.

Solution sketches (dischw4sol.pdf).

1. Prove that $m(n+k)=(mn)+(mk)$ holds for all $m, n, k\in \mathbb N$.

2. Prove that $m(nk)=(mn)k$ holds for all $m, n, k\in \mathbb N$.

3. Prove that $mn=nm$ holds for all $m, n\in \mathbb N$. (Some lemmas will be needed. The lemmas that I found to be useful for this problem prove that (i) $0k=0$, (ii) $1k=k$, and (iii) $(m+n)k=mk+nk$.)

You may use any properties of addition that we have proved.

1. Show that
$f\colon \mathbb N\times \mathbb N\to \mathbb N\colon (m,n)\mapsto 2^m(2n+1)-1$
is a bijection.
(You may use any facts that you know about addition and multiplication of natural numbers, not just the facts proved in class.)

2. Show that if $|X|=|Y|$, then $|{\mathcal P}(X)|=|{\mathcal P}(Y)|$.
(Hint: You must show that if there is a bijection $f\colon X\to Y$, then there must also be a bijection $g\colon {\mathcal P}(X)\to {\mathcal P}(Y)$.)

3. Let $\textrm{Eq}(\mathbb N)$ be the set of equivalence relations on $\mathbb N$. Show that $$|{\mathcal P}(\mathbb N)|\leq |\textrm{Eq}(\mathbb N)|\leq |{\mathcal P}(\mathbb N\times \mathbb N)|.$$ Use the results of Problems 1 and 2 and use the CBS Theorem to conclude that $|\textrm{Eq}(\mathbb N)| = |{\mathcal P}(\mathbb N)|$.
(Hint: For the first part, you should show that there exist injective functions $h\colon {\mathcal P}(\mathbb N)\to \textrm{Eq}(\mathbb N)$ and $k\colon \textrm{Eq}(\mathbb N)\to {\mathcal P}(\mathbb N\times \mathbb N)$.)