CS2013 Tutorial questions on mathematical induction
Aim: to let you get some practice applying mathematical induction to different kinds of problems.
These exercises (which will probably take you a few hours) assume you do know what
mathematical induction
CS 205
Notes on Strings of Symbols
Magnus M. Halldorsson Ann Yasuhara
September 6, 1988
Let 2 be a nite, non-empty set of symbols.
1. Denition of 2
Intuitive idea: We want to dene the set which will contain all possible strings of _
symbols from 2. The no
LECTURE 13
7.0.1
Set operations
A B = cfw_a U |a A or a B (Set union)
A B = cfw_a U |a A and a B (Set intersection)
A B = cfw_a A|a 6 B (Set difference)
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
A = cfw_a U |a 6 A (Set complementation)
LECTURE 8
Examples
There exist two irrational numbers x and y such that xy is rational.
Show that there is a unique r such that ar + b = 0 where a 6= 0
Show that given two rational numbers there is always a rational number
between them
sh
Th
is
ar stu
Example 7 from the textbook
Missing (3,3)
Missing 2,2 , 3,3 , (4,4)
Missing all
Missing all
Since = cfw_1, 2, 3, 4, to be reflexive, must contain 1,1 , 2,2 , 3,3 , (4,4)
So 3 and 5
Example 8 from the textbook
Yes. Because always holds!
No. Because > does
LECTURE 5
Order of quantifiers
Nested different quantifiers represent different statements if they are reversed. So order is important.
Example:
xy(x2 = y)
ar stu
ed d
vi y re
aC s
o
ou urc
rs e
eH w
er as
o.
co
m
Meaning that for every x i the domain the
Review
1
Logic gates:
1 2
2
1
1 + 2
1
2
2
AND
OR
NOT
Boolean circuit construction:
Derive the Boolean function to be computed from the specification
Construct the circuit sequentially
Finite state machine (automaton) as a model of computation
You should b
CS 205 Quiz 01 (09/16/2016) Name: _ NET ID: _
Mm
Problem 1 [30 points] Indicate whether the following statements-are propositions (circle one)
and if so, there truth values (circle one when applicable).
may]
_I@ ' W
_rwmm
m-me-
Problem 2 [20 points] Ne
CS 205 Quiz 02 (09/28/2016) Name: _ _ _ _ _ _ NET ID: _
Problem 1 [20 points] Express the following statements using quantiers7 and then express the
negated form in English (Do not simply use the phrase it is not the case that).
Statement at P(w) Qu
CS 205 Final Coverage
CS 205
Jingjin Yu | Rutgers
Chapter 1
Propositional logic [1.1-1.2]:
Familiar with the basic concepts and definitions, e.g., what is the truth table for basic logic
operations (, , , )
Propositional equivalence, De Morgan's law [1.3
1
b
0
b
b
3
10
b
5
2
b
8
b
b
a
b
b
6
9
b
4
b
b
7
Figure 1: An NFA-
LECTURE 27
12.1
Transitions
In all automata that we have seen so far, every time that it has to change
from one state to another, it must use one input symbol. An NFA- is an
NFA that all
Introduction to 205, and its Sylabus
Why Discrete math: as opposed to continuous.
It is particularly useful for CS, and much of it is useful for programming.
And it will provide a simpler context in which to understand the nature of math, which is
impo
CSCI 241H:
HOMEWORK 6
Show your work.
Prove the following by induction. Show all steps.
1. ni=1 i3 = (n(n + 1)/2)2 for positive integer n.
True for k = 1. Assume for k: ki=1 i3 = (k(k + 1)/2)2 .
Check for k + 1 :
3
3
3
k
k+1
i=1 i = i=1 i + (k + 1) =
(k+1
Math 2200-1. Solutions for Practice Quiz 2. Fall 2008.
Problem 1. The sequence of Lucas numbers is defined recursively by 0 = 2, 1 = 1, and
n = n1 + n2 , for n 2.
(1) Find 8 .
(2) Show by induction that
20 + 21 + + 2n = n n+1 + 2,
whenever n is a nonnegat
CSci 243 Homework 6
Due: 9:00 am, Wednesday, 10/19
My name
1. (Rosen 4.1/12, 8 points) Prove by induction that
Pn
1 i
i=0 ( 2 )
=
2n+1 +(1)n
.
32n
2. (Rosen 4.1/26, 8 points) Suppose that a and b are real numbers with 0 < b < a. Prove by induction
that if
EECS 203
Homework 5 Solutions
Total Points: 40
Page 69:
56)
Suppose that is an invertible function from Y to Z and g is an invertible
function from X to Y. Show that the inverse of the composition f o g is
given by (f o g)-1 = g-1o f 1.
4 points
If a func
CS 205 Sections 07 and 08
Homework 4 Accepted for grading 4/12
1. Prove that whenever p1 , . . . pn is a list of two or more propositions,
(p1 p2 . . . pn )
is logically equivalent to
p1 p2 . . . pn
Use mathematical induction, and the fact that (pq) is eq
CSE 311: Foundations of Computing I
Section: Sets and Modular Arithmetic Solutions
How Many Elements?
For each of these, how many elements are in the set? If the set has infinitely many elements, say so.
(a) A = cfw_1, 2, 3, 2
Solution: 3
(b) B = cfw_, cf
HAND IN
Answers recorded
in question paper
PAGE 1 OF 5 PAGES
QUEEN'S UNIVERSITY
FACULTY OF ARTS AND SCIENCE
SCHOOL OF COMPUTING
CISC-203
DISCRETE MATHEMATICS FOR COMPUTING SCIENCE
TEST 4
November 2005
Professor Selim G. AKL
Please write your answer to eac
CSE 240: Logic and Discrete Mathematics
Release: 02-26-2015
Homework 6
Due: In class (or my mailbox) Thursday, 3-5-2015 1:00pm
Make your solutions concise and formal. Your goal is to convince me that you know the solutions.
You are highly encouraged to ty
CS240
Solutions to Induction Problems
Fall 2009
1. Let P (n) be the statement that n! < nn , where n 2 is an integer.
Basis step: 2! = 2 1 = 2 < 4 = 22
Inductive hypothesis: Assume k! < k k for some k 2.
(We need to show that P (k + 1) is true, given the
LECTURE 4
Example (proving equivalences without using truth tables):
Show that (p (p q) p q
We start from one side: (p (p q)
De Morgan: p (p q)
De Morgan again: p (p q)
Distributive: (p p) (p q)
Negation: F (p q)
Identity: (p q)
1
A tautology is a c
LECTURE 5
Order of quantifiers
Nested different quantifiers represent different statements if they are reversed. So order is important.
Example:
xy(x2 = y)
Meaning that for every x i the domain there is always a value equal to that
number squared. This st
LECTURE 12
6.0.1
Definitions
In the following definitions we will provide a small axiom system for set theory.
A set is a collection of elements. Sets are usually denoted by cfw_set definition.
The definition in the braces can be given in terms of a prop
Rutgers University CS 205 Introduction to Discrete Structures 1 (Summer 2011)
Quiz 7 (due 2011.08.02) (150 points)
Question 1 (50 points total, 5 points each row) For each pair of functions below, determine which
asymptotic relationships hold between them
Rutgers University
CS 205 Introduction to Discrete Structures 1 (Summer 2011)
Quiz 8 (due 2011.08.04) (150 points)
Question 1 (10 points) Prove that (n + 20n) O (n ) by showing constants c and k such that
2
3
=
(c > 0) (n > k ) ( f (n) c g (n) ) for f (n)
Rutgers University
CS 205 Introduction to Discrete Structures 1 (Summer 2011)
Quiz 9 (2011.08.09) (100 points)
Question 1 (20 Points, 5 points each) For each of the graphs below, give the exact number of
strongly connected components and list all nodes in