Generalized Unions/Intersections
A union of a collection of sets is the set that contains all elements that are members of at least one set in the collection.
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An intersection of a collection of sets is the set that contains all elements
Translating Statements
a) A student in your class has a cat, a dog and a ferret.
b) All students in your class have a cat, dog, or ferret.
x(C( x) D( x) F ( x)
x(C( x) D( x) F ( x)
c) Some student in your class has a cat and a ferret, but not a dog.
x(C(
BROCK UNIVERSITY
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Final Examination: December 2009 Course: MATH IP66 Date of examination: 15 December 2009 Time of examination: 09:00 - 12:00
Number of pages: 10 Number of students: 133 Number of hours: 3 hours Instructor: B. Farzad
Examinatio
Section 1.1
Propositional Logic
1
CHAPTER 1
The Foundations: Logic and Proofs
SECTION 1.1
Propositional Logic
2. Propositions must have clearly dened truth values, so a proposition must be a declarative sentence with no
free variables.
a) This is not a pr
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BROCK UNIVERSITY
Page 1 of 11
Final Examination: June 2015 Number of pages: 11
Course: Math IP66 Number of students: 89
Date of Examination: Saturday, June 6, 2015 Number of hours: 3
Time of Ex
Sets
A set is an unordered collection of objects Examples: cfw_apple,orange,pear is a set. cfw_1,2,3,4,5 is a set.
The objects of a set are called the elements (or members) of a set. The set is said to contain its elements.
Notation
= The set of all rea
24) We proceed with a proof by contradiction.
- We assume that it is not the case that "at least 10 days of any 64 days chosen fall on the same
day of the week."
- This means that "less than 10 days of any 64 days chosen fall on the same day of the week."
Page 78
4a) Kangaroos live in Australia and are marsupials. Therefore, Kangaroos are marsupials.
Let the following:
p: Kangaroos live in Australia
q: Kangaroos are marsupials
Therefore we have the following cases:
p q
q
p q
is the statement Kangaroos
live
1
BROCK UNIVERSITY
Page 1 of 10
Final Examination: June 2013
Course: Math 1P66
Date of Examination: Friday. ,June 7, 2013
Time of Examination: 1200-1500
Number of pages: 10
Number of students: 88
Number of hours: 3
Instructor: Basil Nanayakkara
Answer eac
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page 1
Math 1P66 Midterm Examination, Spring 2015
Department of Mathematics & Statistics
Brock University
Answer each question in the space provided. Use the reverse side for rough work. The marks for each
question are shown in the left margi
8) For each of the sets in Exercise 7, determine whether cfw_2 (A set cfw_2) is an element of that set.
a) cfw_x Rx is aninteger greater than1
To explain this set, we have the following elements cfw_2, 3, 4,5, 6, 7,8 , . In this set we have an element 2
MATH IP66
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Review
Recall: Proof methods for a statement p q Direct: Assume p is true, prove q is true. By Contraposition: Assume q is true, prove that p is true. By Contradiction: Assume p is true, assume q is true. If a contradiction is found, the proof is comple
Example
Show that ( p ( p q) and p q are logically equivalent by developing a series of logical equivalences.
( p ( p q) p ( p q) (De Morgan) p [ ( p) q] (De Morgan) p [p q] (double negation) ( p p) ( p q) (distributive) F ( p ( p p q) (negation) (commuta
MATH 1P66 MATHEMATICAL REASONING
Instructor: Jeff Haroutunian
A few notes
Course outline Course materials will be found on Sakai http:/lms.brocku.ca Course text (not required but recommended) is "Discrete Mathematics and Its Applications, 6/e", by Ken
Induction: Review
When using Mathematical Induction to prove that a property P(n) holds for all positive integers n, we must do the following two steps: 1)Basis step: Show that P(1) is true. 2)Demonstrate that if P(k) is true for arbitrary k, then P(k+1
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