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 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

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 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

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

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 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
~
~
. <

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

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 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 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 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

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

PMATH 330 Logic, Solutions to Assignment 6
1. Translate the following statements into formulas in rst-order number theory (after adding the unary function
symbol f ).
(a) x 0 and y x (in the interpretation R).
Solution: Note that this statement is equival

PMATH 330 Mathematical Logic, Solutions to the Midterm
[3]
1: Determine whether the following string X is a formula and, if so, make a derivation for it.
X = (P Q) (R (P Q) (R S )
Solution: This is not a formula. The second open bracket is followed by the

PMATH 330 - Assignment 2
Due Thursday, May 22nd
Please hand in the following problems:
1. Show that the following tables are the basic truth tables for the connectives and , as
we did in class for :
F
T
T
F
F
G
T
F
T
F
(F G)
T
T
T
F
F
T
T
F
F
G
T
F
T
F
(F

PMATH 330 - Assignment 3
Due Thursday, May 29th
Please read Section 2.6 in the course notes and hand in the following problems:
1. Prove that the following statements are true for all formulas F, G and H.
(a) (F G) treq (F G)
(b) (F G) treq (G F )
(c) (F

PMATH 330, Solutions to Assignment 5
In this assignment, let x, y, z, u, v and w be the rst 6 variable symbols, in that order, let a, b and c be
constant symbols, let f , g and h be function symbols with f unary, g binary and h ternary, and let p, q and
r

PMATH 330, Assignment 5
Due Mon July 20
In this assignment, let x, y, z, u, v and w be the rst 6 variable symbols, in that order, let
a, b and c be constant symbols, let f , g and h be function symbols with f unary, g binary
and h ternary, and let p, q an

PMATH 330, Solutions to Assignment 4
1. (a) Let F = P (Q R), G = (QP )R and H = QR. The following is a derivation of the valid
argument F, G | H. Provide justication on each line.
=
Solution: We indicate which rule is used on each line by writing V for a

PMATH 330, Assignment 4
Due Fri July 10
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 |

PMATH 330, Solutions to Assignment 3
1. Let F = (P Q)R (P R) Q .
(a) Use truth-equivalences to nd a DNF formula which is truth-equivalent to F .
Solution: We have
F =
(P Q)R (P R) Q
treq (P Q) R) (P Q) R) (P R) (P R) Q
treq (P Q) R) (P Q R) (P R) (P R) Q

PMATH 330, Assignment 3
Due Mon June 22
1. Let F = (P Q)R (P R) Q .
(a) Use truth-equivalences to nd a DNF formula which is truth-equivalent to F .
(b) Find a CNF formula which is truth-equivalent to F .
(c) Determine whether F is a tautology and determin

PMATH 330, Assignment 2
Due Fri June 5
1. (a) Let F = (P R) (R Q) (P Q). Determine whether | F .
=
(b) Let G = (P R)(QP ) and let H = (Q R)P . Determine whether G treq H.
2. Let F = (QP )R, G = (Q R)S and H = (P R)S. Determine whether
F, G, H is satisable

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