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 $\bigcup {\mathcal P}(x) = x$.

Solution sketches.

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.

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, while if $g\circ f$ is surjective, then $g$ is surjective.
(b) Give an example where $g\circ f$ is injective, but $g$ is not injective, and an example where $g\circ f$ is surjective but $f$ is not surjective.

3. The function $P_A: A\times B\to A: (a,b)\mapsto a$ is called the first projection map, or the projection onto $A$.
(a) What is the image of this function?
(b) What is the coimage of this function?
(Make sure to consider the possibility where $B=\emptyset$.)

Solution sketches.

1. Show that the kernel of a function with domain $A$ is an equivalence relation on $A$.

2. Show that if $E$ is an equivalence relation on $A$, then $E$ is the kernel of some function with domain $A$.
(Hint: You need to find a function with domain $A$ and kernel $E$. Let $P_E = \{[a]_E\;|\;a\in A\}$ be the partition associated to $E$. Show that the natural map $\nu: A\to P_E: a\mapsto [a]$ has kernel $E$.)

3. Suppose that $f: A\to B$ and $g: A\to C$ are two functions with common domain $A$. Let $f\times g$ be the function $A\to B\times C$ be the product function: $a\mapsto (f(a),g(a))$. Show that $\ker(f\times g)=\ker(f)\cap \ker(g)$.

Solution sketches.

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

3. Prove that if $k\in\mathbb N$, then any subset of $k$ is finite.

Solution sketches.

3/20/19

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 took lines from the Book of Ecclesiastes to write a song, which was made famous by the Byrds in 1965. One line is:

To every thing there is a season, and a time for 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 $p$".

Solution sketches.

4/5/19

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

2. This problem also concerns the formal sentence from Problem 1.
(i) Is the sentence true in the natural numbers, $\mathbb N$? Give a winning strategy for the appropriate quantifier.
(ii) Is the sentence true in the real numbers, $\mathbb R$? Give a winning strategy for the appropriate quantifier.
(iii) Negate the sentence, and then rewrite the negation so that it is in prenex form.

3. Show that the following pairs of propositions are logically equivalent. What does each equivalence say about proof strategies?
(i) $H\to C$ versus $(H\wedge (\neg C))\to \bot$.
(ii) $H\to C$ versus $((\neg C)\to (\neg H))$.
(iii) $(H_1\wedge H_2)\to C$ versus $(\neg C)\to ((\neg H_1)\vee (\neg H_2))$.
Solution sketches.

1. Show that ``$A\subseteq B$ and $B\subseteq A$ implies $A=B$'' in each of the following two ways.
(i) With a direct proof.
(ii) With a proof by contradiction.

2. Show that ``any nonconstant, real, linear function $f(x)=ax+b$ has a unique root'' in each of the following two ways.
(i) With a direct proof.
(ii) With a proof by contradiction.

3. The goal of this problem is to prove that the composition of two surjective functions is surjective. 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 variable $a$ range over the set $A$, the variable $b$ range over the set $B$, and the variable $c$ range over the set $C$.

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

To prove that the composition of surjective functions is surjective, 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.

4/19/19

2.
(i) 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?
(ii) 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.

Last One!

2.
(i) How many 4-digit numbers have digits in strictly increasing or strictly decreasing order? (You are counting $abcd$ for $a, b, c, d\in\{0, 1, \ldots,9\}$ such that either $a\lt b\lt c\lt d$ or $a\gt b\gt c\gt d$.)
(ii) How many 4-digit numbers have digits $abcd$ satisfy $a\leq b\leq c\leq d$ or $a\geq b\geq c\geq d$?

3. How many 5-card poker hands have cards of every suit?

Solution sketches.