1. Define $V_0=\emptyset$, $V_1={\mathcal P}(V_0)$, $V_2={\mathcal P}(V_1)$, $V_3={\mathcal P}(V_2)$, and so on.
(a) List the elements of $V_0, V_1, V_2$ and $V_3$.
(b) Draw a directed graph whose ``dots'' are the sets in $V_3$ and where $x\to y$ means $x\in y$. (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),
Solution sketches (video).

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.

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

Solution sketches (dischw2sol.tex),
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.tex),
Solution sketches (dischw3sol.pdf)

1. Prove that $m(n+k)=(mn)+(mk)$ holds for the natural numbers.

2. Prove that $m(nk)=(mn)k$ holds for the natural numbers.

3. Prove that $mn=nm$ holds for the natural numbers. (Some lemmas will be needed.)

Solution sketches (dischw4sol.tex),
Solution sketches (dischw4sol.pdf)

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. (This shows that $|\mathbb N\times \mathbb N|=|\mathbb N|$ using a different argument than the ones given 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 must show that there exist injective functions $h\colon {\mathcal P}(\mathbb N)\to \textrm{Eq}(\mathbb N)$ and $k\colon \textrm{Eq}(\mathbb N)\to {\mathcal P}(\mathbb N\times \mathbb N)$.)

Solution sketches (dischw5sol.tex),
Solution sketches (dischw5sol.pdf)

10/27/21

1. Write the following propositions in disjunctive normal form, assuming that each proposition is a function of $p$, $q$ and $r$.
(i) $p\to r$
(ii) $((p\to q)\to ((\neg p)\leftrightarrow r))$.
(iii) $q$

2. Write the following axioms of set theory as formal sentences.
(i) Extensionality.
(ii) Pairing.
(iii) Power set.

3. In 1959, Pete Seeger wrote a song, Turn! Turn! Turn!, which was made famous by the Byrds in 1965. One line is:

To every thing there is a season, and a time to every purpose under heaven.

Write this as a formal sentence using predicates $S(s,t)$ = "$s$ is the season for thing $t$" and $P(T,p)$ = "$T$ is the time for purpose $p$".
(Note: this example shows that the English language is more flexible than mathematical language in the possible ways to order $\forall$ and $\exists$.)

Solution sketches (dischw6sol.tex),
Solution sketches (dischw6sol.pdf)

1. This problem concerns the formal sentence $$(\forall x)(\forall y)((((\exists z)(x=z^2))\wedge ((\exists z)(y=z^2)))\to ((\exists z)(x+y=z^2))).$$ (a) Draw the formula tree for this sentence.
(b) Standardize the variables apart.
(c) Write the sentence in prenex form.

2. This problem also concerns the formal sentence from Problem 1.
(a) Is the sentence true in the natural numbers, $\mathbb N$? Give a winning strategy for the appropriate quantifier.
(b) Is the sentence true in the real numbers, $\mathbb R$? Give a winning strategy for the appropriate quantifier.

3. Negate the sentence from Problem 1 and then rewrite the negation so that it is in prenex form.
Solution sketches (dischw7sol.tex),
Solution sketches (dischw7sol.pdf)

1. Prove the statement ``If $A\subseteq B$ and $B\subseteq A$, then $A=B$'' in each of the following two ways.
(a) With a direct proof.
(b) With a proof of the contrapositive.

2. Prove the statement ``If $A\cap B=\emptyset$ and $A\cup B=B$, then $A=\emptyset$'' in each of the following two ways.
(a) With a direct proof.
(b) With a proof by contradiction.

3. The goal of this problem is to prove that the composition of two injective functions is injective. The type of structure involved looks like $\mathbb X=\langle A, B, C; f, g\rangle$ where $f:A\to B$ and $g:B\to C$ are functions. Let the variables $a, a'$ range over the set $A$, and the variables $b, b'$ range over the set $B$.

The functions (i) $f$, (ii) $g$, (iii) $g\circ f$ are injective if the following sentences hold in $\mathbb X$:
(i) $(\forall a)(\forall a')((f(a)=f(a'))\to (a=a'))$,
(ii) $(\forall b)(\forall b')((g(b)=g(b'))\to (b=b'))$,
(iii) $(\forall a)(\forall a')(((g\circ f)(a)=(g\circ f)(a'))\to (a=a'))$.

To prove that the composition of injective functions is injective, you must give a winning strategy for $\exists$ in the sentence in (iii). YOU ARE ALLOWED TO USE the fact that there exist winning strategies for $\exists$ in the sentences in (i) and (ii). Write a proof that indicates the winning strategy for $\exists$ in (iii), which accesses the information of the strategies for $\exists$ in (i) and (ii).
Solution sketches (dischw8sol.tex),
Solution sketches (dischw8sol.pdf)

2.
(a) How many paths are there from the point $(0,0)$ of $\mathbb R^2$ to the point $(10,15)$ of $\mathbb R^2$ if each path consists of a sequence of steps of length 1 moving in the direction of the positive $x$-axis or the positive $y$-axis?
(Hint: count paths by counting descriptions of paths, for example strings $xxyxyy\cdots xy$ with $10$ $x$'s and $15$ $y$'s.)
(b) How many paths are there from the point $(0,0,0)$ of $\mathbb R^3$ to the point $(10,15,20)$ of $\mathbb R^3$ if each path consists of a sequence of steps of length 1 moving in the direction of the positive $x$-axis, the positive $y$-axis or the positive $z$-axis?

3. Let $MC(n,k)$ be the number ``$n$-multichoose-$k$''. Use a combinatorial argument to show that
$MC(n,0)+MC(n,1)+\cdots+MC(n,k)=MC(n+1,k)$.

Solution sketches (dischw9sol.tex),
Solution sketches (dischw9sol.pdf)

Last One!

2.
(a) How many binary relations on the set $X=\{x_1,x_2,\ldots,x_n\}$ are there?
(b) How many binary relations on $X$ are reflexive?
(c) How many binary relations on $X$ are reflexive and symmetric?
(d) Explain why there are $B_n$ binary relations on $X$ that are reflexive, symmetric, and transitive.

3. These problems are about seating people at a round table. Two seating arrangements are considered the same if they differ by a rotation. (So, for example, the arrangement ABCDEF is the same as BCDEFA.)
(a) How many ways are there to seat 3 couples at a round table?
(b) What if couples must sit together?
(c) What if couples are not allowed to sit together?

Solution sketches (dischw10sol.tex),
Solution sketches (dischw10sol.pdf)