Proving By Induction Natural Numbers
Prove by induction that for all natural numbers x >= 2 and n >= 0,
x^n - 1 is divisible by x - 1.
. Let S(n) to be "for all natural numbers x >= 2, x^n - 1 is divi
B W3 2 " W
C Qkxohr AX
Skim36x29 ImYmve. our SEAR in
n Cmgumkvg frames (f?3 our knowbdaa
in leak) I amok
o ffemk 1>r.
We. also learn More numb. Mum-Q fads.
n A MOEHMCJ 3% (Err, TJAS: G cafa%U\3
f-QSI
CMPUT 272 Fall 2017
Assignment 2
Posted Thurs Sept 21
Due Thurs October 5 at 10:45AM in your seminar sections drop box across from CSC 145
This is the mandatory assignment cover sheet;
without it, you
CMPUT 272 Fall 2017
Exercises for Seminar #4: Oct. 2/3
These are the seminars practice questions. The TAs will go over the solutions to these and other
questions in the seminars.
1. (Epp 3.3.45) The n
Week 5: Proofs
Agenda:
Direct proofs
More example of proof techniques
Operations mod n and an application
Reading:
Textbook: 4.3-4.4, 4.6-4.7
1
Week 5: Proofs
General steps in such a proof:
Make
CMPUT 272 Fall 2017
Solutions to Exercises for Seminar #4: Oct. 2/3
These are the seminars practice questions. The TAs will go over the solutions to these and other
questions in the seminars.
1. (Epp
Using Crowdmark to Submit your Written Assignments
All written assignments for this course will be administered through Crowdmark. Crowdmark is a
web-based platform for the online marking of written w
Week 2: Logic
Agenda:
2.3 Valid and Invalid Arguments
2.4 Digital circuits.
Reading:
Textbook 2.3-2.4
1
Arguments vs Statements:
Statement: a declarative sentence that is either true or false, but
Week 3: Logic
Agenda:
2.4-2.5 Circuits and number systems
3.1-3.3 Predicate and Quantified Statements
1
Week 3: Logic
Design circuits for addition
how do we add single digit numbers?
how to take
Week 4: Predicate logic
Agenda:
More on predicate logic
Direct proofs
Reading:
Textbook Sections 3.1-3.4, 4.1-4.2
1
Relations between quantifiers and negation:
Predicate p(x), its negation is p(x)
Rules of Interference
Statement: a declarative sentence that is either true or false, but
not both
{ E.g., \The sum of x and y is greater than 0" is NOT a statement
Argument: a sequence of statements
Truth Test Questions
We say a statement form is satisfiable if there is a
truth assignment to its variable such that the whole statment
becomes true. Therefore, a formula that is not satisfiable is al
More examples of logical equivalences:
Example 1: It is easy to see that:
9x(P(x) ^ q(x) ) 9x P(x) ^ 9x Q(x)
but these are not equivalent; for example let P(x) be 2x + 1 = 5
and Q(x) be x2 = 9.
Exampl
Some Common Errors:
There is a gap between the basis and induction step.
We \prove" the following predicate for all n 1:
Given any set of n 1 di
erent lines in the plane no two of which
are parallel t
Applications to Cryptograph and Proofs
Theorem: p2 is irrational.
Proof: By way of contradiction, assume that p2 is rational, i.e.
there are integers a; b 6= 0 such that p2 = a
b
.
Without loss of gen
Prove by induction that for all natural numbers n >= 0:
n
sum i^2 = n(n + 1)(2n + 1)/6
i=1
n
. Let S(n) be " sum i^2 = n(n + 1)(2n + 1)/6".
i=1
. We prove that S(n) holds for all integers n >= 1.
, Ba
Propositional Logic (Overview):
\A collection of rules for deductive reasoning that are intended to
serve as a basis for the study of every branch of knowledge."
The concept of argument form (versus i
Coefficient Questions
Question: What is the coefficient of x^2y^3z^3 in (x+y+z)^8?
(x+y+z)^8=(x+y+z)(x+y+z).(x+y+z) (8 times).
The coefficient of x^2y^3z^3 is the number of ways we can
select x from 2
Elementary Number Theory
First we will see one more example about the relational database system
of a library we were discussing last week. Let's consider the following
query:
\Find the names of all s
UNIVERSITY OF ALBERTA
CMPUT 272
Term Test 2, Nov 2016
Duration: 75 min.
Practice Test
Aids allowed: One single sheet of notes (double sided). Nothing else.
Total Pages = 8
Page 1
contd. . .
Last Name:
UNIVERSITY OF ALBERTA
CMPUT 272
Sample Term Test 1, Oct 2016
Duration: 70 min.
Last Name:
First Name:
Do not turn this page until you have received the signal to start.
(In the meantime, please fill o
Week 1: Introduction
Agenda:
Welcome to CMPUT 272
Official course information
Logical form and logical equivalence
Reading:
Introductino to the course and policies
textbook: sections 2.1-2.2
1
D
Gorilla Mourguefiles.org
LincolnPark-01b.jpg
1.With ape-like fury (Pg. 16)Compares Hydes fury with that of
an apes, gives a vivid description in
few words.
2.Where the lamps glimmered like
carbuncles