Ve203 Discrete Mathematics
Assignment 6
Date Due: 4:00 PM, Thursday, the 5th of November 2015
Exercise 6.1. The sums of the digits of numbers can be used to obtain a variety of results about the numbers:
i) Show that a positive integer is divisible by 3 i
Ve203 Discrete Mathematics
Assignment 8
Date Due: 4:00 PM, Thursday, the 19th of November 2015
Exercise 8.1. Show that if k, n N with 1 k n, then
( )
n
nk
k1
k
2
(2 Marks)
Exercise 8.2. Show that if A and B are events and P is a probability function, the
Ve203 Discrete Mathematics
Assignment 10
Date Due: 4:00 PM, Thursday, the 3rd of December 2015
Exercise 10.1. Determine whether each given graph is planar. Either draw an isomorphic graph without
crossing edges, or prove that the graph is non-planar.
u4
u
Ve203 Discrete Mathematics
Assignment 12
Date Due: 4:00 PM, Tuesday, the 15th of December 2015
Exercise 12.1. Use induction in (N N \ cfw_(0, 0), ), the set of pairs of natural numbers with lexicographic
ordering induced by the ordering of N \ cfw_0, to s
Ve203 Discrete Mathematics
Assignment 4
Date Due: 4:00 PM, Thursday, the 22nd of October 2015
Exercise 4.1. Prove Corollary 1.6.11 of the lecture:
Let a, b Z with |a| + |b| = 0. Then
T (a, b) = cfw_n Z : n = ax + by, x, y Z
is the set of all integer multi
Ve203 Discrete Mathematics
Assignment 9
Date Due: 4:00 PM, Thursday, the 26th of November 2015
Exercise 9.1. In the following graphs, nd the number of vertices, the number of edges and the degree of each
vertex. Identify all isolated and pendant vertices.
Ve203 Discrete Mathematics
Assignment 7
Date Due: 4:00 PM, Thursday, the 12th of November 2015
Exercise 7.1. Prove that in a bit string, the string 01 occurs at most one more time than the string 10.
(2 Marks)
Exercise 7.2. Consider the scheme for countin
Ve203 Discrete Mathematics
Assignment 11
Date Due: 4:00 PM, Thursday, the 10th of December 2015
Exercise 11.1. Complete the IDEA survey for Ve203.
(5 Bonus Marks)
Exercise 11.2. Show that a full m-ary balanced tree of height h has more than mh1 leaves. De
Handout 3
Natural Deduction for Predicate Logic
EECS 203
Fall 2017
This handout contains Natural Deduction rules of inference for the predicate calculus.
If we have a predicate formula , we can make another formula [xt] by substituting the term t for all
Handout 2
Natural Deduction Examples
EECS 203
Fall 2017
Before we give examples of Natural Deduction proofs, we will discuss why we use
proof systems and, in particular, why we chose this proof system.
It seems easy to determine whether a proposition foll
EECS 203: Discrete Mathematics
Fall 2017
Discussion 1 Notes
1
Definitions
Proposition:
Consistent System :
Converse:
Contrapositive:
Inverse:
Tautology:
Contradiction:
Logically equivalent ():
1. Exercise 1.1.15
Let p, q, and r be the propositions
Handout 1
Natural Deduction Rules
EECS 203
Fall 2017
Handouts 1 through 4 supplement the material in section 1.6 of the textbook. This
handout presents the Natural Deduction rules of inference for propositional logic.
(-intro)
(-intro)
p q
pq
(-elim)
p
pq
Handout 4
Predicate Logic Proofs
EECS 203
Fall 2017
This handout contains the examples of Natural Deduction proofs for predicate logic.
The rules are in Handout 3. There is an introduction and elimination rule for the
quantifiers and , and for the equalit
Ve203 Discrete Mathematics
Assignment 3
Date Due: 4:00 PM, Thursday, the 15th of October 2014
Exercise 1. For this exercise, you may use everything you know about complex numbers from calculus.
i) Show that the set S = cfw_z C : |z| = 1 is a group (S, ) w
Ve203 Discrete Mathematics
Assignment 5
Date Due: 4:00 PM, Thursday, the 29th of October 2015
Exercise 5.1. Use Fermats Little Theorem to compute 52003 mod 7, 52003 mod 11 and 52003 mod 13. Then use
the Chinese Remainder Theorem to compute 52003 mod 1001
Ve203 Discrete Mathematics
Assignment 2
Date Due: 6:00 PM, Thursday, the 27th of September 2012
Exercise 1. Fuzzy sets are dened as sets S = cfw_x : P (x) where for any x the statement P (x) is assigned a
truth value as in Exercise 7 of Assignment 1. This
Ve203 Discrete Mathematics
Assignment 11
Date Due: 6:00 PM, Thursday, the 6th of December 2012
Exercise 1. In the following graphs, nd the number of vertices, the number of edges and the degree of each
vertex. Identify all isolated and pendant vertices. C
Ve203 Discrete Mathematics
Assignment 9
Date Due: 6:00 PM, Thursday, the 22nd of November 2012
Exercise 1. For the following questions it is recommended that you set up a generating function for the answer
and then use a computer to nd the corresponding c
Ve203 Discrete Mathematics
Assignment 7
Date Due: 6:00 PM, Thursday, the 8th of November 2012
Exercise 1. Let M, N be nite sets with card M = card N and M N . Prove that M = N .
(2 Marks)
Exercise 2. Use the Pigeonhole Principle or Theorem 2.2.16 of the l
Ve203 Discrete Mathematics
Assignment 3
Date Due: 6:00 PM, Thursday, the 11th of October 2012
Exercise 1. Use strong induction to show that every n N \ cfw_0 can be written as a sum of distinct powers
of 2, i.e., as a sum of a subset of integers 20 = 1, 2
Ve203 Discrete Mathematics
Assignment 8
Date Due: 6:00 PM, Thursday, the 15th of November 2012
Exercise 1. Show that the Fibonacci numbers satisfy the recurrence relation fn = 5fn4 + 3fn5 for n =
5, 6, 7, . . . together with the initial conditions f0 = 0,
Ve203 Discrete Mathematics
Assignment 5
Date Due: 6:00 PM, Thursday, the 25th of October 2012
Exercise 1. What sequence of pseudorandom numbers is generated using the linear congruential generator
xn+1 = (4xn + 1) mod 7 with seed x0 = 3?
(2 Marks)
Exercis
Ve203 Discrete Mathematics
Assignment 12
Date Due: 6:00 PM, Thursday, the 13th of December 2012
Exercise 1. Determine whether each given graphs is planar. Either draw an isomorphic graph without crossing
edges, or prove that the graph is non-planar.
u4
u3
Ve203 Discrete Mathematics
Assignment 1
Date Due: 6:00 PM, Thursday, the 20th of September 2012
You are required to compose your solutions in neat and legible handwriting. Up to 10% of the total score may
be deducted solely due to the apearance and legibi
Ve203 Discrete Mathematics
Assignment 6
Date Due: 6:00 PM, Thursday, the 1st of November 2012
Exercise 1. Show that if a and m are relatively prime positive integers, then the inverse of a modulo m is
unique modulo m.
(3 Marks)
Exercise 2. The goal of thi
Ve203 Discrete Mathematics
Assignment 4
Date Due: 6:00 PM, Thursday, the 18th of October 2012
Exercise 1. Order the letters M,I,C,H,I,G,A,N alphabetically using
i) merge sort,
ii) insertion sort,
iii) bubble sort
algorithms. (Note that it does not matter
Ve203 Review Class I
By Wang Tianyu
Basic Concepts in Logic
Negation
Conjunction
Disjunction
Basic Concepts in Logic
De Morgan rules
Basic Concepts in Logic
Implication
Equivalenc
e
Basic Concepts in Logic
Contraposition
Basic Concepts in Logic
Example
Ba