Math 201 Khoi Nguyen
Section 1.6
Problem 1
Find the argument form for the following argument and determine
whether it is valid. Can we conclude that the conclusion is true if the
premises are true?
If Socrates is human, then Socrates is mortal.
Socrates i

Math 201 Khoi Nguyen
Page 187
Problems 5
Show that if A and B are sets, then A (A B) = A B.
Proof:
First:
Let x A (A B) x A and x
A B.
Because, x A B x A and x
A or x B
B. Thus, x
Therefore, x A and (x
Because x A, so x
and x B
A is impossible, so x B. Th

Lectures 8-9
Practice Problems
CSI30
1. Let universe U = cfw_1,2,3,4,5,6,7,8,9,a,b,c,d,e,f, and sets A = cfw_a, b, c, d, 1, 2, 3, 4,
B = cfw_1,2,3,4,5, and C = cfw_a,b,1,2,9. Find
a) A U B U C
b) A B C
c) A B
d) C A
e) A U B
f) A C
g) A U B C
1
Practice P

Discrete Mathematics: Solutions to Homework 2
1. (12%) For each of the following sets, determine whether cfw_2 is an element of that
set.
(a) cfw_x R| x is an integer greater than 1
(b) cfw_x R| x is the square of an integer
(c) cfw_2,cfw_2
(d) cfw_2,cfw_

Introduction to Discrete math
nal review
1. Use a truth table to show that P (Q R) R (Q P )
2. Prove the equivalence from problem 1 without using a truth
table.
3. Use a direct proof to show that if d = gcd(a, b) and a = bq + r
then d is a common divisor

Introduction to Discrete math
Unit 4
exam
1. State the Division Algorithm and use it to find q and r for each
of the following pairs of a and b
(a)
(b)
(c)
(d)
a = 200, b = 7
a = 5, b = 12
a = 0, b = 13
a = 13, b = 0
2. Extending the Division Algorithm to

Answers to Proofs:
Math 2534
Answers to Proof Homework sheet.
1) The sum of any even integer and any odd integer is odd.
Proof:
Let a be the even integer and b be the odd integer. By definition of even and odd we
have that a = 2n and b = 2m + 1. Consider

Direct proof : A => B
Contraposition : indirect proof : not B => not A
Contradiction : A ^ not B > contradiction
If and only if : prove A => B ; B => A
I II III : prove by I => II ; II => III ; III => I - or - I II and
I III
Without loss of generality l t

Khoi Nguyen- math 201
Section 1.1
Problem 12
a) p q
If you have the flu, then you will miss the final
examination.
b) q r
You dont miss the final examination if and only if
you pass the course
c) q r
If you miss the final examination, then you wont
pass t

Math 55, Solutions to In-class Problems
Feb 12, 2013
1. Problem: prove that if a b mod m where a, b, m Z and m 2
then gcd(a, m) = gcd(b, m).
Solution: We will show that gcd(b, m) divides gcd(a, m). By symmetry, we will have also shown that gcd(a, m) divid

Math 201 Khoi Nguyen
Section 1.7
Problem 17
Show that if n is an integer and n3 + 5 is odd, then n is even using
a) a proof by contraposition.
Let n is odd integer
By definition of odd integer: a Z, n = 2a + 1
n3 + 5 = (2a + 1)3 + 5 = 8a3 + 3. 4a2. 1 + 3.

V
P => Q ( inverse is P => Q ) ( converse is Q => P )
(contrapositive Q => P)
: null
Cardinality => s lng elements
Power set > th l cfw_ none, cfw_x ,cfw_y , cfw_x,y
: union
: intersection
A
B
C
Membership > ex: divisor of 15 > cfw_-15,-5 ,. ,5 ,15

ICS 141: Discrete Mathematics I (Fall 2014)
4.1 Divisibility and Modular Arithmetic
Divides
a | b means a divides b. That is, there exists an integer c such that b = ac. If a | b, then b/a is an
integer.
If a does not divide b, we write a6 | b.
Properties

Introduction to discrete math
Unit I test review
1. Use a truth table to discover which of the following are equivalent.
B (A C)
[A (B C)]
(A C) B
1
2. Prove what you discovered in problem 1 without using a truth
table.
3. Rewrite the following sentenc

Math 201 Khoi Nguyen
REVIEW TEST 1
1.
A
B
C
T
T
T
T
T
F
T
F
T
T
F
F
F
T
T
F
T
F
F
F
T
F
F
F
A
T
T
T
T
F
F
F
F
B
T
T
F
F
T
T
F
F
A
T
T
B
T
T
C
T
F
T
F
T
F
T
F
C
T
F
AC
T
F
T
F
F
F
F
F
(B C)
F
T
T
T
F
T
T
T
B (A C)
T
F
T
T
F
F
T
T
A V (B C)
T
T
T
T
F
T
T
T

V > hoc | ^ > v
What is the minimum number of students, each of whom comes
from one of the 50 states, who must be enrolled in a university to
guarantee that there are at least 100 who come from the same state?
The minimum number of students is the smalles

ICS 141: Discrete Mathematics I (Fall 2014)
1.6 Rules of Inference
An Inference Rule is a pattern establishing that if we know that a set of premise statements of
certain forms are all true, then we can validly deduce that a certain related conclusion sta

Math 201 Khoi Nguyen
Section 1.7
Problems 2
Use a direct proof to show that the sum of two even integers is even
Let m and n be the even integers
By definition of even integer: a, b Z, m = 2a and n = 2b
m + n = 2a + 2b = 2(a + b)
m + n is even
The sum o

Math 201 Khoi Nguyen
Section 1.8
Problem 6
Prove using the notion of without loss of generality that 5x + 5y is
an odd integer when x and y are integers of opposite parity.
Without loss of generality, suppose that x is odd integer and y is
even integer
By

Khoi Nguyen Math 201
Section 1.1
Problem 28
a) If it snows tonight, then I will stay at home.
Converse: If I stay at home, then it snows tonight
Contrapositive: If I dont stay at home, then it doesnt snow.
Inverse: If it doesnt snow tonight, then I dont s

Khoi Nguyen Math 201
Section 2.2
Problem 2
Suppose that A is the set of sophomores at your school and B is the
set of students in discrete mathematics at your school. Express each
of these sets in terms of A and B.
a) the set of sophomores taking discrete

Math 3322 Test II
DeMaio Fall 2009
Name
Instructions. Show all your work. Credit cannot and will not be awarded for work not shown. Where
appropriate, simplify all answers to a single decimal expansion.
1. (10 points) A certain shirt is produced for both

Homework # 9 Solutions
Math 111, Fall 2014
Instructor: Dr. Doreen De Leon
Determine whether or not each of the statements is true or false. Prove your assertion.
1. Suppose A, B, and C are sets. If A B, then A C B C.
Solution: This statement is true.
Proo

CS 2336
Discrete Mathematics
Lecture 3
Logic: Rules of Inference
1
Outline
Mathematical Argument
Rules of Inference
2
Argument
In mathematics, an argument is a sequence of
propositions (called premises) followed by a
proposition (called conclusion)
A

Another probability definition
The probability of an event is the number of favorable cases
divided by the total number of equally likely cases
If 10 coins fall to the floor what is the probability that 5 are
heads and 5 are tails.
si= (a1,a2,a3,a4,a5,a6,

FAKULTI SAINS KOMPUTER DAN MATEMATIK
CSC 510 - Discrete Structures
Assignment 1
10. For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain
the rules of inference used to obtain each conclusion from the premises.

Homework 7
3/18/2015
SOLUTIONS
In-class Exercise 19.
(a) Consider strings of length 10 consisting of 1s, 2s, and/or 3s.
(i) How many of these are there with no additional restrictions?
Answer: 31 0 (three choices for each digit)
(ii) How many of these are

Rosen 4.5 p342.
1. In how many ways can five elements be selected in order from a set with three elements when repetition is
allowed? Perm with repetition; each of five selections has three choices : 35 = 243.
3. How many strings of six letters are there?

ICS 141: Discrete Mathematics I (Fall 2014)
2.1 Sets
A set is an unordered collection of objects, called elements or members of the set. A set is said to
contain its elements.
We write a A to denote that a is an element of the set A. The notation a
/ A d