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.)

HW1 Problems (dischw1.tex),
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.
(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 $\emptyset\times A = \emptyset$.

HW2 Problems (dischw2.tex)
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.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.)

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

Remark: On Friday, Feb 14, we discussed all ideas needed to solve these three problems. BUT, we will prove all the laws of addition on Monday Feb 17, so you might want to wait until then before trying these problems. I think that it might help to see similar examples worked out before you start on these.

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.
(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 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.pdf)

1. Determine whether the negation of the proposition $P$ logically implies, is logically equivalent to, or is logically independent of proposition $Q$:
(i) $P= a\to b$, $Q = a\wedge (\neg b)$.
(ii) $P=(a\to b)\to a$, $Q=\neg a$.
(iii) $P=(a\to b)\wedge (b\to c)$, $Q=(a\to c)$

2. 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) $q$

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

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.

Latex code for the tree in Problem 1! (Formula_Trees.tex)
Latex code for the tree in Problem 1! (Formula_Trees.pdf) (You do not have to use Latex. It is enough to draw images by hand and upload those with your work.)

Solution sketches (dischw7sol.pdf)

All proofs should be informal proofs!

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 structures 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.pdf)

1. This problem involves a deck of 52 distinct playing cards.
(a) In how many ways can a 13-card bridge hand be dealt from the deck?
(b) How many different 13-card bridge hands are there?
(Parts (a) and (b) are different, because cards are dealt in an order, while hands are unordered.)

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.pdf)

2. Let $X=\{x_1,x_2,\ldots,x_n\}$.
(a) How many binary relations on $X$ 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 seating arrangement ABCDEF is the same seating arrangement 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.pdf)