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
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
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 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 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
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.
MATH1090
Problem Set No. 2
October 2015
Lassonde Faculty of Engineering
EECS
MATH1090. Problem Set No. 2
Posted: Oct. 7, 2015
Due: Oct. 27, 2015, by 2:00pm; in the course
assignment box.
It is worth remembering (from the course outline):
The homework must
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
MATH1090
Problem Set No 3- Solutions
November 2012
Faculty of Science and Engineering
MATH1090- Solutions to Problem Set No 3
1. (3 marks) Prove A B
A B.
It is easy to see that A B
for A being and B being
A B since the tautological implication does not ho
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
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
York University
Department of Electrical Engineering and Computer Science
Lassonde School of Engineering
MATH1090A. Mid Term Test, of October 21, 2015
Solutions
Posted October 23, 2015
Question 1. (a) (1 MARK) Through truth tables or related short cuts sh
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) .
(
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 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
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
1
Math 1090 A
Summer 2009
Answers to Test 2
Each question is worth 10 marks.
1. Prove Contradiction: A A .
Hint: This can be done with a 4-step Equational proof, using Axiom
(10) (Golden Rule).
Proof 1
A A
Axiom (10) (Golden Rule)
A A A A
Axiom (9) (Exclu
[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
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 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
Mathematical Induction
Section 5.1
(Review)
Learning Outcomes
Use Mathematical induction to prove
recurrence relations, summation formulae and
inequalities.
Climbing an
Infinite Ladder
Suppose we have an infinite ladder:
1. We can reach the first rung of
MATH 1090 A
Assignment 3
Page 1
due: June 28, 2017
1. (11 points) Consider the following string:
(x)f(x, y) = x)(y)g(x, y) = f(x, y) (x)f(x, y) = x)(y)g(x, y) = f(x, y)
(a) Prove this string is a first order formula (well-formed formula) using bottom-up o
MATH 1090 Midterm Practice
Questions Made/Compiled by Gian Alix
Introduction to Logic for Computer Science
June 15, 2017
This is meant to test your knowledge, not as a guide on what to focus on.
Also note that these questions are meant to be tougher for p
SC/MATH 1090
4.1 The First-order Language
of Predicate Logic
4.2 Axioms and Rules of First-order Logic
Ref: G. Tourlakis, Mathema'cal Logic, John Wiley & Sons, 2008.
Adapted from slides by Dr. Vida Movahedi
York University
Department of MathemaV