PMATH 330 Mathematical Logic, Course Outline, Spring 2015
Lectures: MWF 2:30-3:20, in MC 1056.
Instructor: Stephen New, oce MC 5419, extension 35554, oce hours WF 12:00-2:00.
Text: Lecture notes will be provided.
Course Outline: We will cover all of the m

ASSIGNMENT 6: DUE NOVEMBER 20
(1) Let L = cfw_0, 1, +, , < be the language of number theory, and N, Z, Q, and R be the
usual interpretations with underlying sets, N, Z, Q, and R, respectively.
(a) Determine, for each sentence, whether or not it is true in

ASSIGNMENT 5: DUE NOVEMBER 10
(1) Suppose f , g, and h are function symbols with f unary, g binary, and h ternary. Let
a, b, and c be constants, and x, y, z be variables. Convert the following terms to bracket
notation:
(a) hgabf xz
(b) f gahgxf yzf gcx
(

ASSIGNMENT 5 SOLUTIONS
(1) Suppose f , g, and h are function symbols with f unary, g binary, and h ternary. Let
a, b, and c be constants, and x, y, z be variables. Convert the following terms to bracket
notation:
(a) hgabf xz
(b) f gahgxf yzf gcx
(c) ghf

ASSIGNMENT 7: DUE DECEMBER 1
(1) Let F and G be first order formulas.
(a) Show that F 9xF .
(b) Show that 8x (F ^ G) treq (8x F ^ 8x G)
(c) Give a language, formulas F and G, an interpretation, and an assignment to show
that it is not necessarily true tha

REVIEW ASSIGNMENT
As the title of this assignment indicates, it is for review. It shouldnt be submitted because it
wont be graded. I also wont be posting detailed (or possibly any) solutions. They are practices
problems to remind you of the stu we did all

ASSIGNMENT 2: SOLUTIONS
(1) Find a truth table for (P ^ Q) $ S) _ (R ^ P ).
Solution: I will strive to not give you any truth tables to fill out that are any larger
than this.
P
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
Q
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
R
1
1
0
0
1

ASSIGNMENT 1: SOLUTIONS
(1) Which of the following are statements? You can give a short explanation if
you are not confident in your choice, but no explanation is necessary to get
marks.
(a) My cat knows how to drive a car.
(b) Do you like pickles?
(c) If

ASSIGNMENT 3 SOLUTIONS
You can use any named rules form the lecture notes that are posted online, but be
sure to refer to anything you use. You should always justify your answers, even if it
doesnt explicitly say to. There appear to be many proofs on this

ASSIGNMENT 4: DUE OCTOBER 30
Do not be alarmed. This assignment is not as long as it looks.
This assignment is more computation heavy than proof heavy. Please try to present your
work in an organized, legible way. While I think its a good idea to do a rou

ASSIGNMENT 4: DUE OCTOBER 30
Do not be alarmed. This assignment is not as long as it looks.
This assignment is more computation heavy than proof heavy. Please try to present your
work in an organized, legible way. While I think its a good idea to do a rou

ASSIGNMENT 6 SOLUTIONS
(1) Let L = cfw_0, 1, +, , < be the language of number theory, and N, Z, Q, and R be the
usual interpretations with underlying sets, N, Z, Q, and R, respectively.
(a) Determine, for each sentence, whether or not it is true in each o

PMATH 330 Introduction to Mathematical Logic
Lecture Notes
by Stephen New
0
Chapter 1. Propositional Formulas
1.1 Denition: The alphabet (or symbol set) of propositional logic consists of the
following symbols:
, , , , , (, ), P1 , P2 , P3 ,
The symbols

PMATH 330, Assignment 7
Not to be handed in
In this assignment, let x, y and z be variable symbols, let c be a constant symbol, let f and
g be function symbols with f unary and g binary, and let r be a binary relation symbol.
In questions 1 and 2, let V b

PMATH 330, Assignment 6
Due Mon July 27
1. Translate each of the following statements about integers into formulas in rst-order number theory.
(a) |x| |y|.
(b) x is squarefree (that is x has no perfect square factors other than 1)
(c) y is positive and z

PMATH 330, Assignment 4
Due Mon July 13
1. (a) Let F = P (QR), G = (QP )R and H = QR. The following is a derivation
of the valid argument F, G | H. Provide justication on each line.
=
1.
cfw_P, QP, R, Q | Q
=
2.
cfw_P, QP, R, Q | QP
=
3.
cfw_P, QP, R, Q |

GgCKAQK g RQv/ 800%
Assignment 10
1. Perform the indicated substitutions.
(a) [$1 + $2]?
a" >63 3(- 041 Grudt -. szW 300%
Assignment 10
(b) [figg$1$g]%: (here f is 3~ary function symbol, 9 is a unary func
tion symbol, and c is a constant symbol)
S5LC5

Groom: QQwLN 350%
Assignment 12
1. Let E x cfw_j g, R: S? c where f 18 a unary function symbol, 9 is a binary
function symbol, R is a binary relation symbol, 8 is a ternary relation
symboh and c is a constant symbol Show that each of the following
E-for

fad Q 1M ZMVWX Sin
Assigm'nont 1 1
1. Let E m cfw_[3 R? S, a where f is a, unary function symboll R is a binary
relation symbol, 3 is a. ternary relation symbol, and c is a constant
syi'nbol. Let :2:, y, and 3 (lei'iote distinct variables. Put the follo

ASSIGNMENT 7: DUE DECEMBER 1
(1) Let F and G be a first order formulas.
(a) Show that F 9xF .
(b) Show that 8x (F ^ G) treq (8x F ^ 8x G)
(c) Give a language, an interpretation, and an assignment to show that it is not
necessarily true that
9x (F ^ G) tre

NOTES ON EQUIVALENCE RELATIONS
Here is a quick introduction to equivalence relations:
Example 1. Lets say, for integers, x and y that x y if and only if x y is a
multiple of 7. Here are a few things to observe:
(1) For any x 2 S, x x. This is because x x

ASSIGNMENT 2: DUE OCTOBER 6
(1) Find a truth table for (P ^ Q) $ S) _ (R ^ P ).
(2) Let F be a formula. Is cfw_F, F satisfiable?
(3) (a) Find truth tables for P $ Q, (P _ R) ! Q, and R ! (P ^ Q). You
may use one truth table that includes a column for eac

ASSIGNMENT 3: DUE OCTOBER 16
There appear to be many proofs on this assignment, and I suppose there are. Most
of them should not be long to write up, but you may need to think about them and
use things that were done in class. If you are stuck for too lon

PMATH 330
Assignment 3
Fall 2013
Due Monday, October 7, 2013 by 11:20 a.m.
1. In this problem we let = cfw_p0 , p1 , p2 , . . .. We assign distinct odd positive integers (s) to symbols
of LP( ) (except parentheses) as follows:
s
(s)
1
3
5
7
9 11
p0
13
p1

PMATH 330
Assignment 5
Fall 2013
Due Wednesday, October 23, 2013 by 11:20 a.m.
1. A student submitted the following incomplete Natural Deduction proof:
( )
( )
( ( )
Add labels/slashes to turn this into a correct Natural Deduction proof of cfw_ |= ( (

PMATH 330
Assignment 3
Fall 2013
Due Monday, October 7, 2013 by 11:20 a.m.
1. In this problem we let = cfw_p0 , p1 , p2 , . . .. We assign distinct odd positive integers (s) to symbols
of LP( ) (except parentheses) as follows:
s
(s)
1
3
5
7
9 11
p0
13
p1

PMATH 330
Assignment 2 revised
Fall 2013
Due Friday, September 27, 2013 by 11:20 a.m.
1. Give the parsing tree for each of the following formulas.
(a) (p (q (r s).
(b) (p2 (p1 p0 ) (p2 ).
(c) (p3 p7 ) (p1 p2 ) (p2 (p5 ).
2. Give the formula which correspo

PMATH 330
Assignment 1
Fall 2013
Due Friday, September 20, 2013 by 11:20 a.m.
1. By comparing truth tables, prove that (x y ) (x y ) = (x y ).
Proof.
xy
0
1
1
1
xy
00
01
10
11
xy
0
0
0
1
(x y ) (x y )
1
0
0
1
xy
0
1
1
0
(x y )
1
0
0
1
By comparison, we se