Math 2090 Homework 4
Solutions
1. Prove
` S T T 6= .
Answer: If S T , T must contain an element which is not in S, so in particular, T contains
an element. I know four different proofs. This one is by Contrapositive and uses the Deduction
Theorem.
We prov
Math 2090 Homework 3 due March 11 at Midnight
Solutions
1. Prove (14.13),
` S T S U S 6= T U .
Answer: Choose w, u to be fresh variables.
S T S U S 6= T U
(3.65)
=
S=
6 (S T S U T U )
Definition of
=
(u  : u S) (S T S U T U ) .
so by (9.30) it suffices
York University
Faculty of Arts, Faculty of Science
Math 2090 N
Midterm Test 2
SOLUTIONS
Instructions:
1. There are 4 questions on 4 pages.
2. Answer all questions.
3. Your work must justify the answer you give.
Question
1
2
3
4
Total
Points
8
8
7
7
30
Ma
York University
Faculty of Arts, Faculty of Science. Atkinson College
Math 2090
Midterm Test
SOLUTIONS
Instructions:
1. Time allowed : 75 minutes.
2. There are 6 questions on 7 pages. Page 3 is
blank.
3. Answer all questions.
4. Prove, means prove using t
York University
Faculty of Arts, Faculty of Science
Math 2090 N
Midterm Test 1
SOLUTIONS
Instructions:
1. There are 4 questions.
2. Answer all questions.
3. Your work must justify the answer you give.
Question
1
2
3
4
Total
Points
6
8
8
8
30
Marks
MATH 20
York University
Faculty of Arts, Faculty of Science
Math 2090 P
Midterm Test 2
SOLUTIONS
Instructions:
1. There are 4 questions on 4 pages.
2. Answer all questions.
3. Your work must justify the answer you give.
Question
1
2
3
4
Total
Points
8
8
7
7
30
Ma
York University
Faculty of Arts, Faculty of Science
Math 2090 P
Midterm Test 1
SOLUTIONS
Instructions:
1. There are 4 questions.
2. Answer all questions.
3. Your work must justify the answer you give.
Question
1
2
3
4
Total
Points
8
8
6
8
30
Marks
MATH 20
Math 2090 Homework 2
Solutions
These are discrete mathematics application questions. The theme is telescoping sums (and products).
1. (a) Prove ` (+j2 j n : a[j 1]) = (+j1 j n 1 : a[j]).
Note: Do not use (8.22). Review the proof steps for (8.22) given i
Math 2090 N
Quiz 2 Version 1 January 30, 2003
SOLUTIONS
1. (7 points) Prove that,
` cfw_z, y cfw_x, y z = x z = y .
Answer: Take w to be a fresh variable.
=
=
=
=
=
=
=
cfw_z, y cfw_x, y
(11.13), Abbreviation
(w  w cfw_w  w = z w = y : w cfw_w  w = x w
8.4.1
(b) Ans: False. Counter example: Let \Gamma be cfw_ A, A and \alpha be A.
Then both \alpha and \alpha are theorems of \Gamma.
(d) Ans: True. Suppose \Gamma is not PDconsistent, then there exist \delta
such that both \delta and \delta are PD deri
You will be expected to know all the rules of SD, how to write down properly the
justification for any of the steps allowed in an SDderivation, etc. Whiteley and
Ganong both feel that you should have done enough derivations in SD by now (three
dozen? fiv
Some Logic Problems
The following problems are adapted from a book of Logic Problems by Raymond
Smullyan: The Lady or the Tiger? and Other Logic Puzzles, Times Books, 1982. Last
updated 02/02/97
Solution of these problems does involve applications of the
General Information
Instructors
Walter Whiteley
email address: whiteley@mathstat.yorku.ca
Office S616 Ross. Phone 33971
Office hours: 9:30 10:00 MWF (in Curtis LH D), and by appointment (see me before or after
class or email me).
Class MWF 8:30, Curtis
Due Wednesday February 12 in the Assignment Box on the 5th floor of North Ross by
4 p.m., or in class Wednesday by 2:30 p.m., i.e., the beginning of class.
1. 5.5.3 f. on p. 215  give a printout from SYMLOG.
I don't remember if I said in class on 5 Febru
Mathematics 2090.03 Whiteley's Assignment 2:
Due Monday January 27 in the Assignment Box on NRoss 5th floor by 12:00 noon.
From our text SYMLOG
Page 96: Exercise 3.1.6. [Use W and B for the two atomic sentences!]
Page 107: Exercise 3.3.3 (h) Print the w
Translating English to SL (Chapter 2 and Section 3.1)
There are a few general comments to keep in mind when translating between natural
language and the formal languages we have.
Other Formal Translations of Language
Overall warning: This translation to S
The assignment for our class is different from that for Whiteley's in several respects.
So go by this, not by his list of questions.
1. 5.3.3 b.
(I did not say this in class on 29 January, but a SYMLOG printout is preferable
here.)
2. 5.4.1 b.
Note that t
Solutions to Whiteley's Test I
Question 1: 12 marks
SD+
++
 PREMISE
 1
A v (B & C)


 PREMISE
 2
D > B


 PREMISE
 3
(A v H)


 DM3
 4
A & H


 &EL4
 5
A


 &ER4
 6
H


 DS5,1
 7
B & C


 &EL7
 8
B


 DN8
 9
B
Solutions to Assignment 7
9.3.16. UD:=cfw_1,2 Rxx:= cfw_(1,1), (2,2), (1,2), (2,1) It is obvious that the first
sentence is satisfied in this model. Also, the second sentence is satisfied:
whenever you have x and y such that Rxy is true (for example x=1,
<HTML>
<HEAD>
<TITLE>Whiteley's suggested problems
</TITLE>
</HEAD>
<BODY>
<h3> Suggested Problems </h3>
The list may include some questions which also appear
on assignments. <br>
Last updated 3/03/97<p>
Problems from Chapter 9
<ul>
<li> Section 9.2 Page