Name (print):
Signature:
ID Number:
PMATH 330, Introduction to Mathematical Logic
University of Waterloo
Final Examination, Fall Term, 2007
Instructor: Stephen New
Date:
December 17, 2007
Time:
7:30-10:00 pm
Instructions:
Question
1. Place your name, sign

PMATH 330 - Ast 6 Solutions
1. (i) Consider the following set of clauses:
cfw_Q, T cfw_P, Q cfw_Q, S cfw_P, R cfw_P, R, S cfw_Q, S, T cfw_P, S, T cfw_Q, S cfw_Q, R, T
Here we go with DPP:
We cannot do any clean up here so lets proceed to resolution

PMATH 330 Assignment IV due Oct 7th
This document will get longer after each lecture!
Answer each problem with justification, except where indicated. That means writing
actual sentences.
1. Let = cfw_P (Q R),
Q (R P ),
P (Q R).
Find a set of formulas whic

Assignment 7
Page 1 of 18
~c1w, . Ho~" ~
1. Let L
~
I 0 'f" -'
Due: June 21, 11:59pm
.
= cfw_f, g, R, 5, c, d wnere f is a unary function symbol , g is a
binary function symbol, R is a binary relation symbol, 5 is a ternary
relation symbol, and c an

Assignment 12 Page 1 of 11 Due: July 26, 11:59pm
Mi Mi Huang
1. Let [I = cfw_f, g, R, S, c Where f is a unary function symbol, 9 is a binary
function symbol, R is a binary relation symbol, S is a ternary relation
symbol, and c is a constant symbol. Show t

These notes are largely based on the notes of Stephen New, which can be found
at www.uwaterloo.ca/snew. Incidentally, it seems that his notes are based on the
course notes of Peter Hoffmann, which are also available online. This course is meant
to be a ge

PMATH 330 NOTES ON ISOMORPHISMS
Definition 1. Let A and B be L-structures for some language L, with domains A and B,
respectively.
An isomorphism from A to B is a function J : A B such that
(i.) J is a bijection
(ii.) For each n-ary relation symbol R of L

Axioms for Propositional Logic
For any propositional formulas , , and , the following formulas will be axioms
of our proof system:
(1) ( ( )
(2) ( ( ) ( ) ( )
(3) ( ) ( )

Axioms for First Order Logic
For any L-formulas , , and , any variables x and y, and any L-term t, the
following are basic axioms of our proof system:
(1) ( ( )
(2) ( ( ) ( ) ( )
(3) ( ) ( )
(4) (x xt ) if t is substitutable for x in
(5) (x ( ) (x x )
(6

Name (print):
Signature:
ID Number:
PMATH 330, Introduction to Mathematical Logic
University of Waterloo
Final Examination, Winter Term, 2005
Instructor: Stephen New
Date:
April 14, 2005
Time:
2:00-5:00 pm
Instructions:
Question
1. Place your name, signat

Assignment 9
Page 1 of 18
Due: July 5, 1l:59pm
G-ru&.aA ", \-\ ()(\~:; l~
1. Translate the following statements about the natural numbers into for
mulas in the language of arithmetic, LaT' You may use any formulas
or abbreviated formulas introduced in

Assignment 5
Due: June 7, 1l:59pm
Page 1 of 15
~: \-\on~d; \-v.u~
1. For each of the following deriv~ ons, provide the justifications for each
line of the derivation.
(a) For any formulas <p and , , vve have cfw_. <p f- (<p
(1) (.'If;
(2) ('<p
~
~
. <

PMATH 330 Assignment V due Oct 21st (note the
date)
This document will get longer after each lecture!
Answer each problem with justification, except where indicated. That means writing
actual sentences.
1. (a) Suppose ` and ` . Show that is not satisfiabl

PMATH 330 Assignment I due Sept 16th
This document will get longer after each lecture!
September 21, 2016
Answer each problem with justification.
1. Consider the sentence if it rains today, then it rains today. Under what circumstances
is this sentence tr

PMATH 330 ASSIGNMENT I DUE SEPT 16TH
THIS DOCUMENT WILL GET LONGER AFTER EACH LECTURE!
Answer each problem with justification.
(1) Consider the sentence if it rains today, then it rains today. Under what circumstances is this sentence true? Under what cir

PMATH 330 Assignment VIII due Nov 25th
This document will get longer after each lecture!
Answer each problem with justification, except where indicated. That means writing
actual sentences.
1. Let L = cfw_+, 0, where + is a binary function symbol, and 0 i

Page 1 of 15
Assignment 6
Due: June 14, 1l:59pm
G-sudD.A- l~dl ~
1. vVhich of the following sets of formulas are consistent? Justify your
answers, using anything up to and including all the Week 6 cont ent .
(a) cfw_ (P
-7
\'" to '" S \~-\e.
Cd) , (Q

Assignment 3
Page 1 of 13
Due: May 24, 11:59pm
6-ro&M" "Io.'V'\ '?o.' I ne.
1. (a) Show that cfw_P, Q, -.(P !\Q) is an unsatisfiable3-element set, each
of whose 2-element subsets is satisfiable.
s u.rr ~
eCry":, Q~
i-tev\ 4>
'S
Y)cfw_)
\-
~
-c:
('So
e

Assignment 4
Due: May 31, 11:59pm
Page 1 of 12
u-rad.oA TCtn
<.
Po.yn.
1. Use the algorithm described on page 9 of the Week 4 Slides to decide
whethpr ear h of the following sets of Horn formulas are satisfiable.
(a) cfw_P, -,(8 1\ Q), (P 1\ H) ~ .).' ),

Assignment 8
Page 1 of 17
Due: June 28, 11:59pm
G-[t.~ . LC\V"\ J=>o.yne
1. (a) Prove part (2) of the Lemma on page 3 of the Week 8 Slides. That
is, show that if cp and 'ljJ are formulas of a first order language
M~
and I = (A. J) is an interpretatio

Assignment 10
Page 1 of 17
6-r~', \-\-O~6l ~Q~
1. Perform the indicated substitutions,
Due: July 12 , 11:59pm
Assignment 10
Page 2 of 17
~ . \-\O~: \4wA~
(b) [J X2g Xl X2]; (here f is 3-ary function
Due: July 12, 11:59pm
symbol, 9 is a unary func
tion

Assignment 11 Page 1 of 12 ' Due: July 19, 11:59pm
1. Let E = cfw_ f , R, S, c Where f is a unary function symbol, R is a binary
relation symbol, S is a ternary relation symbol, and c is a constant
symbol. Let-1:, y, and 2 denote distinct variables. Put t

PMATH 330 Logic, Solutions to Assignment 3
1. (a) Let F = (P Q) R). By making a truth-table, nd a DNF formula and a CNF formula, both of
which are truth-equivalent to F .
Solution: We make a truth-table.
P
Q
R
1
1
1
1
0
0
0
0
1
1
0
0
1
1
0
0
1
0
1
0
1
0
1

PMATH 330, Solutions to Assignment 1
1. For each of the following strings, determine whether the string is a formula. If so, provide a derivation with
justication at each step. If not, then explain why not.
(a) (P Q)(R Q) P )
Solution: This string is not