MATH 251-3, Fall 2007
Simon Fraser University
Assignment 4: Solutions
Additional Question Due: 4:30pm, Monday 15 October 2007
1. The position of a particle at time t is given by r(t) = 2 t, t2 , t3 3 .
(a) Find the velocity v, speed v = |v| and accelerati
MATH 251-3, Fall 2007
Simon Fraser University
Assignment 1: Solutions
Additional Question
Due: 4:30pm, Monday 17 September 2007
1. Describe clearly in words the geometry of the surface or region in R3 described by
the graph of the following equation or in
MACM 101 Discrete Mathematics I Exercises on Propositional Logic. Due: Tuesday, September 29th (at the beginning of the class)
SOLUTIONS 1. Construct a truth table for the following compound proposition: (p q ) (p q ) Solution p q (p q ) (p q ) 00 0 01 1
10-2
Discrete Mathematics Theorems and Proofs
Previous Lecture
Axioms and theorems
Rules of inference for quantified statements
Rule of universal specification
Rule of universal generalization
Existential rules
Theorems and Proofs II
Introduction
Discrete
4-2
Discrete Mathematics Laws of Logic
Previous Lecture
Truth tables
Tautologies and contradictions
Logic equivalences
Laws of Logic
Introduction
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics Laws of Logic
4-3
Laws of Logic
4-4
Discrete Mathema
Problems to Week 3 Tutorial MACM101 (Fall 2016)
1. Use truth tables to verify absorption law.
2. Show that (p q) (q r) (p r) is a tautology (use equivalences).
3. Show that the following compound statements are logically equivalent:
(p r) (q r) and (p q)
Problems to Week 4 Tutorial MACM 101 (Fall 2016)
1. Consider the universe of all polygons with three or four sides, and
define the following predicates for this universe:
A(x):
E(x):
H(x):
P (x):
Q(x):
R(x):
S(x):
T (x):
all interior angles of x are equal
Problems to Week 5 Tutorial MACM 101 (Fall 2016)
1. Determine which of the following arguments are valid and which are invalid. Provide an
explanation for each answer. (Let the universe consist of all people presently residing in
Canada.)
(a) All mail car
2-2
Discrete Mathematics Propositional Logic
What is Logic?
Computer science is a mere continuation of logic by other means
Georg Gottlob
Propositional Logic
Introduction
Contrariwise, continued Tweedledee, if it was so, it might be; and if it
were so, it
6-2
Discrete Mathematics - Logic Inference
Previous Lecture
Rules of substitution
Logic inference
Inference and tautologies
Rules of inference
Conjunctive Normal Form
Resolution
Predicates and Quantifiers
Introduction
Discrete Mathematics
Andrei Bulatov
D
8-2
Discrete Mathematics Predicates and Quantifiers II
Previous Lecture
Predicates
Assigning values, universe, truth values
Predicates
and Quantifiers II
Introduction
Discrete Mathematics
Andrei Bulatov
8-3
Discrete Mathematics Predicates and Quantifiers
12-2
Discrete Mathematics Operations on Sets
Venn Diagrams
Often it is convenient to visualize various relations between sets.
We use Venn diagrams for that.
Operations on Sets
Introduction
set
universe
B is a subset of A
B
A
Discrete Mathematics
Andrei B
MACM 101 Discrete Mathematics I
Exercises on Propositional Logic. Due: Tuesday, September 27th (at the beginning of the class)
Reminder: the work you submit must be your own. Any collaboration and
consulting outside resources must be explicitly mentioned
MACM 101 Discrete Mathematics I
Exercises on Predicates and Quantifiers.
Due: Tuesday, October 11th (at the beginning
of the class)
Reminder: the work you submit must be your own. Any collaboration and consulting outside resources must be explicitly menti
MACM 101 Discrete Mathematics I Exercises on Functions and Induction. Due: Tuesday, November 10th (at the beginning of the class)
Reminder: the work you submit must be your own. Any collaboration and consulting outside resourses must be explicitely mentio
MACM 101 Last Name
This is a sample!
First Name and Initials
Midterm Test 2
Some Day, 2007
Student No. NO AIDS allowed. Answer ALL questions on the test paper. Use backs of sheets for scratch work. Total Marks: 100 1. Give a definition of the codomain of
Final Exam on
MACM-101
Discrete Mathematics
1. What is an open variable?
2. Prove that sets A and B are disjoint if and only if A ∪ B = A∆B.
3. If f ◦ g is one-to-one, does it follow that g is one-to-one?
4. What does it mean that a function f is in O(g)
MATH 251-3, Fall 2007
Simon Fraser University
Assignment 10: Solutions
Additional Question Due: 4:30pm, Wednesday 5 December 2007
1. Let E be the part of the sphere of radius a in the first octant; that is, x2 +y 2 +z 2 = a2 , x 0, y 0, z 0. (a) The volum
MATH 251-3, Fall 2007
Simon Fraser University
Assignment 9: Solutions
Additional Question
Due: 4:30pm, Monday 26 November 2007
1. Sketch the region of integration, and evaluate the following integral:
4
0
2
√
√
y cos(x4 ) + x sin y dxdy
y
Solution:
√
The
MATH 251-3, Fall 2007
Simon Fraser University
Assignment 8: Solutions
Additional Question Due: 4:30pm, Monday 19 November 2007
1. Find the point(s) on the surface x2 y 4 z = 8 that is/are closest to the origin. Solution: We should immediately begin by for
MATH 251-3, Fall 2007
Simon Fraser University
Assignment 7: Solutions
Additional Question
Due: 4:30pm, Monday 5 November 2007
1. In this problem, you will show that if f (x, y) is harmonic (that is, satisﬁes Laplace’s
equation), then z(x, y) = f (x2 − y 2