Math 1090 M Homework 1
Answers and Solutions
1. p, , (p q) are Boolean formulae.
p is a symbol from A1,
is a symbol from A2, and as p and q are Boolean formulae
by property III, so is (p q).
(p), , p q are not Boolean formulae.
2. p, , (p ), is not a form
Math 1090 A
Summer 2012
Answer to Homework 3
Exercise from Section 1.5 (p. 50)
15. Calculate the rst bullet below using Denition 1.3.15, step-by-step.
For each of the remaining bullets, if its dened, just write the result;
if its not dened, explain why.
Math 1090 A
Summer 2012
Answers to Homework 5
Exercises from Section 3.6 (pp. 108109)
8. Use Resolution to prove A (B C) A B.
Proof
A (B C) A B
if, by the Deduction Theorem,
A (B C)
AB
if, by 2.6.7 (Proof by Contradiction),
A (B C), (A B)
if, by 2.4.23 (1
Math 1090 A
Summer 2012
Homework 4
Exercises from Section 2.7 (pp. 8687): 13, 612, 14, 16, 1820
Instructions
Exercise 8 asks you not to use certain metatheorems. In the other exercises,
you may use any axiom, theorem or metatheorem on the list that I post
SC/MATH 1090
2.2-2.5 Equational Proofs
Ref: G. Tourlakis, Mathematical Logic, John Wiley & Sons, 2008.
Template of slides: Vida Movahedi
York University
Department of Electrical Engineering and Computer Science
York University- MATH 1090
05-Equational
1
L
Math 1090 M Homework 3
Answers and Solutions
1. Tourlakis 2.7:
1. Prove
A B A B A .
A B A B
< Axiom (8) >
A (B B)
< Axiom (4), Leib, C-part: A p, p fresh >
A
< (2.4.10) >
A
2. Prove
A (A B) A .
A (A B)
< Axiom (10) >
AAB AAB
< Axiom (7), Leib, C-part
[Type text]
SC/MATH 1090
1.1- Boolean
Formulae
Ref: G. Tourlakis, Mathematical Logic, John Wiley & Sons, 2008.
Template of slides: Vida Movahedi
York University
Department of Electrical Engineering and
Computer Science
York University- MATH 1090
01-Boolea
York University
Lassonde School of Engineering
Midterm Test
Math 1090 A
NAME (print):
(Family)
(Given)
SIGNATURE:
STUDENT NUMBER:
Instructions:
1. There are 8 questions on 8 pages.
2. Answer all questions.
3. Your work must justify the answer you give.
Qu
Math 1090 M Homework 7
Answers and Solutions
Tourlakis 8.3:
1. Take D = cfw_0, 1, A to be x = 0, and B to be x = 1. Then (x)A is f, (x)B is f, so that
(x)A (x)B is t. But (x)(A B) is f.
This proves that the schema
(x)A (x)B) (x)(A B)
is not universally va
MATH1090
Problem Set No1
September 2011
Faculty of Science and Engineering
MATH1090. Problem Set No1
Posted: Sept. 20, 2011
Due: Oct. 4, 2011, by 2:00pm; in the course
assignment box.
It is worth remembering (from the course outline):
The homework must be
Math 1090 A
Summer 2012
Answers to Homework 6
Exercise 3 from Section 4.3 (p. 149)
Let f be a unary function, g a binary function, 7 a constant, and <
a binary predicate, which we write between its arguments, instead of
before them.
For each of the foll
Math 1090 A
Summer 2012
Answers to Homework 4
Exercises from Section 2.7 (pp. 8687)
1. Prove A B A B A.
Proof
This is Theorem 2.4.12.
2. Prove Absorption 1: A (A B) A.
Hint: This can be done with a 3-step Equational proof, starting with
a use of Axiom (10
Math 1090 M Homework 6
Answers and Solutions
Tourlakis 6.6:
8.
(x)(B A C)
< WL, (2.4.11), C-part (x)p >
(x)(B A C)
< (6.4.2), x dnof in A >
A (x)(B C)
< WL, (2.4.11), C-part A (x)p >
A (x)(B C)
9. In standard notation, this is
A (x)(B C) (x)(B A C) .
(
3. (3 marks) Prove that the last symbol of a Boolean formula is never the symbol :. Hint.
Use analysis of formula calculation, or prove by induction on formulae.
Proof
We use induction on formulas to prove that, for every Boolean formula A, the statement
Math 1090 A
Summer 2012
Answers to Homework 8
Exercises from Section 8.3 (pp. 208210)
1. Provide a countermodel to show that the converse of Ax3,
(x)A (x)B (x)(A B),
()
is not a universally valid schema.
Conclude that () is not a theorem schema.
Answer
Co
Math 1090 A
Summer 2012
Assignment 2
I. Section 2.7 (pp. 8687): Exercises 4, 5, 17
Instructions
In your proofs, you may use anything on the theorem list for Boolean Logic
that I posted, except for 3.2.1 (Posts Theorem). Some of the hints below
give additi
SC/MATH 1090
1.2- Induction on the Complexity of
WFF: Some Easy Properties of WFF
Ref: G. Tourlakis, Mathematical Logic, John Wiley & Sons, 2008.
Template of slides: Vida Movahedi
York University
Department of Electrical Engineering and
Computer Science
Y
MATH 1090
Problem Set #4
March/April 2006
York University
Faculties of Science and Engineering, Arts, Atkinson
MATH 1090. Problem Set #4
Posted 28 March 2006
Due: 6 April 2006, 4:00pm,
in my box (5th oor, North Ross)
PLEASE NOTE THERE ARE TWO PAGES
Sectio
Math 1090 A
Summer 2012
Assignment 1
I. Exercises from Section 1.5 (pp. 4849)
5. Use induction on formulas to prove the desired property for formulas
that have not been abbreviated by omitting brackets.
8. Use induction on formulas to prove the desired pr
Math 1090 A
Summer 2012
Answers to Homework 1
Exercises from Section 1.5 (pp. 4849)
In the following exercises, we assume that formulas are unabbreviated.
4. Use induction on formulas to prove that the string () is not a Boolean
formula.
Proof
We use indu
SC/MATH 1090
Introduction to
Logic for Computer Science
York University
Department of Electrical Engineering
and Computer Science
York University- MATH 1090
00-MATH1090
1
Mathematical Logic What?
Prerequisites: SC/MATH 1190 3.00 or SC/MATH 1019 3.00
Tru
SC/MATH 1090 A INTRODUCTION TO LOGIC FOR COMPUTER SCIENCE
Department of Electrical Engineering and Computer Science
T/R, 10 11:30 a.m., Life Science Building 106
Fall 2013 Course Outline
COURSE DIRECTOR: Natasha May, Lassonde 2018, Office Hours: T/R 11:30
Math 1090 A
Summer 2009
Answers to Assignment 1
Exercises from Section 1.5 (pp. 4849)
5. Use induction on formulas to prove that the string (
formula. Assume that formulas are unabbreviated.
) is not a
In this answer, formula means unabbreviated formula.
Math 1090 Sample Midterm Solutions
1.
p
q
r
(p)
(p) q)
(r p)
(p) q) (r p)
2.
p
p
A
F (A) =
A
(F (B)
(F (B) F (C)
if
if
if
if
if
if
A=p
A=p
A = q for some other propositional variable q
A = or A =
A = (B)
A = (B C) for some cfw_, , ,
Proof by inducti
York University
Department of Computer Science and Engineering
Faculty of Science and Engineering
MATH1090 A. Mid Term Test, October 20, 2009Solutions
Professor George Tourlakis
Question 1. (a) (1 MARK) Through truth tables or related short cuts show that
MATH1090
Problem Set No. 1 Solutions
September 2016
Lassonde Faculty of Engineering
EECS
MATH1090A. Problem Set No1 Solutions
Posted: Oct. 5, 2016, by 3:00pm
The concept of late assignments does not exist in this course.
1. (3 MARKS) Prove that no wff end
MATH1090A
Problem Set No. 3 Solutions
November 2016
Lassonde School of Engineering
EECS
MATH1090. Problem Set No. 3
Posted: Nov. 16, 2016,
It is worth remembering (from the course outline):
The homework must be each individuals own work. While consultatio
MATH1090A. Mid Term Test, October 26, 2016 Solutions
Question 1.
The questions on this page are semantic. Syntactic proofs will not be accepted !
(a) (2 MARKS) Through truth tables or related short cuts show that
(A) B |=taut (A B)
Proof. Pick a state v t
MATH1090
Problem Set No. 2 Solutions
October 2016
Lassonde Faculty of Engineering
EECS
MATH1090. Problem Set No. 2 SOLUTIONS
Posted: Oct. 21, 2016
It is worth remembering (from the course outline):
The homework must be each individuals own work. While con