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Math 2001001: Intro to Discrete Math, Spring 2019


Homework




Assignment

Assigned

Due

Problems

HW0 
1/16/19

1/23/19

Read pages 18, 2024.

HW1 
1/23/19

1/30/19

Read pages 2527.
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.

HW2 
1/31/19

2/6/19

1. Show that if $A, B$ and $C$ are sets,
then $\{A, B, C\}$ is a set in each of the following ways:
(i) Using the Axiom of Replacement.
(ii) Without using the Axiom of Replacement.
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.

HW3 
2/7/19

2/13/19

Read
Sections 2.2 and 2.3 (Functions and binary relations, pages 2845).
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.

HW4 
2/13/19

2/20/19

Read pages 4751.
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.

HW5 
2/21/19

2/27/19

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

HW6 
3/8/19

New Due Date!
3/20/19

Read Subsections 3.1, 3.5.1, 3.6.1, and Subsections 4.1.14.1.3.
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.

HW7 
3/20/19

New Due Date!
4/5/19

Read Subsections 4.2.1, 4.2.3, 4.3.2.
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.

HW8 
4/5/19

4/10/19

All proofs should be informal proofs!
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.

HW9 
4/10/19

New Due Date!
4/19/19

1. This problem involves a deck of 52 distinct playing cards.
(i) In how many ways can a 13 card bridge hand be dealt from the deck?
(ii) How many different 13 card bridge hands are there?
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.

HW10
Last One! 
4/19/19

4/24/19

1. You have just given birth to octuplets.
How many ways can you name your children
if you only like the names Billy Bob, Jim Bob and Sue Bob?
(Any child is allowed to receive any of these three names.)
2.
(i) How many 4digit 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 4digit numbers have digits $abcd$
satisfy $a\leq b\leq c\leq d$ or $a\geq b\geq c\geq d$?
3. How many 5card poker hands have cards of every suit?
Solution sketches.


