p (q p)
Axiom, Identity of :
true
Reexivity of :
Metatheorem:
(3.3)
(3.4)
(3.5)
(3.7)
Distributivity of over :
Contradiction:
Absorption:
(3.41)
(3.42)
(3.43)
De Morgan:
(3.47)
Axiom, Distributivity of over :
Axiom, Denition of :
p q p q
Double negation:
Ladies or Tigers: The Eighth Trial
"There are no signs above the doors!" exclaimed the prisoner. "Quite true", said the king. "The signs were just made, and I haven't had time to put them up yet." "Then how do you expect me to choose?" demanded the prison
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 4
COMP SCI 1FC3 Mathematics for Computing
27 January 2010
Everything on this sheet is within scope for Midterm 1. Exercises will be discussed not only in
the tutorial
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 3
COMP SCI 1FC3 Mathematics for Computing
20 January 2010
In addition to your handwritten paper submission, hand in at least four calculational proofs in solution
A
t
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 6
COMP SCI 1FC3 Mathematics for Computing
10 February 2011
Exercise 6.1 (Textook p. 176)
Dene suitable predicates and functions and then formalize the sentences that
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 2
COMP SCI 1FC3 Mathematics for Computing
13 January 2010
A
Hand in your solution to one of the Assignment Questions (your choice) electronically as L TEX document
(a
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 5
COMP SCI 1FC3 Mathematics for Computing
3 February 2011
Exercise 5.1 (Textook p. 175)
Translate the following English statements into predicate logic.
(a) Some inte
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 1
COMP SCI 1FC3 Mathematics for Computing
6 January 2010
Exercise 1.1
Translate the following sentences into Boolean expressions. You may choose to give names to the
Ladies or Tigers: First and Third Cases (Ex. 3.9)
In the first case, the following signs are on the doors of the rooms: 1 In this room there is a lady, and in the other room there is a tiger. 2 In one of these rooms there is a lady, and in one of these ro
McMaster University Department of Computing and Software Dr. W. Kahl
COMP SCI 1FC3 Sheet 3
COMP SCI 1FC3 - Mathematics for Computing
19 January 2012 Exercise 2.8' Write truth tables to compute values for the following expressions in all states. (a) b c d
McMaster University Department of Computing and Software Dr. W. Kahl
COMP SCI 1FC3 Sheet 4
COMP SCI 1FC3 - Mathematics for Computing
26 January 2012 Exercise 4.1 Prove Identity of (3.30), p false p, by transforming its more structured side into its simple
McMaster University Department of Computing and Software Dr. W. Kahl
COMP SCI 1FC3 Sheet 5
COMP SCI 1FC3 - Mathematics for Computing
2 February 2012 Exercise 5.1 Translate the following English statements into predicate logic. (a) n is greater than 23. (b
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 7
COMP SCI 1FC3 Mathematics for Computing
26 February 2011
Exercise 6.7 building on (3.83) Axiom, Leibniz: (e = f ) (E [z := e] E [z := f ])
Prove the following theor
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 11
COMP SCI 1FC3 Mathematics for Computing
20 March 2011
Exercise 11.1 (Midterm 2 Question 3)
When proving numbered theorems, use only earlier theorems. If you apply
Cheat Sheet
p true = p !
p q = q p!
(p q) r = p (q r)!
p (q r) = (p q) (p r)!
p p = false!
p false = p !
p q = q p!
(p q) r = p (q r)!
p (q r) = (p q) (p r)!
p p = true!
(unit)
(symmetry)
(associativity)
(distributivity)
(complement)
p p = p!
p false = fa
CS 1FC3 Logic Notes
Musa Al-hassy
<alhassy@gmail.com>
April 6, 2011
pi
=1
Contents
1 Terminology
2
2 Inferences
3
3 English to Predicate Logic
3
4 Flaws
4
5 Proof Techniques
5
Introductory remarks
1. These notes are not meant to be read in a linear fashio
Outline
COMP SCI 1FC3
Goal:
Conscious and precise use of this language is the foundation for precise specication and rigorous reasoning, which take a central place in
this course.
To a large degree, this can be seen as analogous to acquiring language skil
Introduction
A
L TEXBasics
Mathematics
Bibliography
Slides
Resources
A
Introduction to LTEX
Christian Kascha
January 21, 2005
Christian Kascha
A
Introduction to L TEX
Introduction
A
L TEXBasics
Mathematics
Bibliography
Slides
Resources
Basic Document Stru
CalcCheck Manual
(Covering CalcCheck-0.2.9 and CalcStyleV6)
Wolfram Kahl
4 April 2011
Abstract
A
This document describes the use of the prototype proof checker CalcCheck and the accompanying L TEX
package CalcStyle to be used for presenting and checking p
The Not So Short
A
Introduction to L TEX 2
A
Or LTEX 2 in 174 minutes
by Tobias Oetiker
Hubert Partl, Irene Hyna and Elisabeth Schlegl
Version 5.00, December 14, 2010
ii
Copyright 1995-2010 Tobias Oetiker and Contributors. All rights reserved.
This docume
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 9
COMP SCI 1FC3 Mathematics for Computing
12 March 2011
Exercise 9.1
Translate the following English statements into predicate logic (using the language of set theory
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 8
COMP SCI 1FC3 Mathematics for Computing
6 March 2011
Exercise 8.1
(a) Prove the generalised De Morgan theorems (9.18a,b,c)
( x R P )
( x R P )
( x R P )
(b) Prove t
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 10
COMP SCI 1FC3 Mathematics for Computing
13 March 2011
This is a Bonus Assignment. Solutions are due Thursday, 24th March, electronically
as CalcCheck-checked sourc
McMaster University
Department of Computing and Software
Dr. W. Kahl
COMP SCI 1FC3
Sheet 12
COMP SCI 1FC3 Mathematics for Computing
26 March 2011
Exercise 11.4
(14.3) Axiom, Cross product:
Prove the following:
(a) (14.4)
Membership:
(b) (14.5)
x, y S T
bS