Predicate Logic & Quantication,
Inferences & Proofs
EECS 203: Discrete Mathematics
Lecture 3 Winter 2013
From Last Lecture
A truth table for quantiers
x P(x)
True P(x) is true for every x
when: in the domain of discourse
x P(x)
P(x) is true for at lea
EECS 203: DISCRETE MATHEMATICS Homework 8 Solutions
1. (4 points) A family has thirteen children: 5 identical quintuplets, two sets of 3 identical triplets, and 2 identical twins. How many distinguishable ways are there to seat the thirteen kids at a roun
EECS 203: Homework 8 Solutions
Section 5.4 1. (E) 8 By binomial theorem, the coefficient of the term x8 y 9 in the expansion of (x + y)17 is 17 = 24310. 8 When considering the term x8 y 9 in the expansion of (3x + 2y)17 , let x = 3x and y = 2y, meaning th
Discrete Mathematics
Assignment 8
Date Due: 8:00 PM, Thursday, the 14th of July 2011
Oce hours: Tuesdays, 1:003:00 PM, and Wednesdays, 12:001:00 PM
Exercise 1. Let (an ) be a sequence of real numbers. We dene the sequences of backward dierences (k an )
EECS 203, Discrete Mathematics Your last name (print):
Winter 2010, University of Michigan, Ann Arbor Your rst name (print):
Circle your lecture section: 1 (TTH9) 2 (TTH12) Circle your discussion section: 011
(Mandava M1:30)
012
(Wu F3:30)
013
(Wu W10:30)
EECS 203: DISCRETE MATHEMATICS Homework 9 Solutions 1. (9 points) If R V V is a binary relation over V then R1 (the inverse) is dened as: (x, y ) R if and only if (y, x) R1 . Prove or disprove the following: (a) If R is transitive & reexive then R R1 is a
EECS 203: Homework 1 Solutions
Section 1.1 1. (E) 8bef b) You do not miss the final exam if and only if you pass the course. e) If you have the flu then you do not pass the course, or if you miss the final examination then you do not pass the course. f) Y
EECS 203
Fall 2014 Exam 2
PRACTICE EXAM A
Problem 1.
Zero or more of the following statements are true. Which ones are they?
(a) The number of positive integers less than 1,000,000 that have the sum of their digits
equal to 9 is 10 .
6
Answer: False. This
EECS 203, Discrete Mathematics
Winter 2007, University of Michigan, Ann Arbor
Problem Set 5
Problems from the Textbook
9.1: 24[E] 9.2: 58[M] 9.3: 32[M], 68[M] 9.4: 16[E] 2.4: 26[C], 38[E] *
Additional Problems Required for All Students
Problem A5.
EECS 203: DISCRETE MATHEMATICS Homework 7 Solutions 1. (4 points) There are 15 gold coins, all of which are identical except for one counterfeit coin, which is either slightly heavier or slightly lighter than all the rest. You have at your disposal a bala
Practice Final Solutions
EECS 203
Winter 2015
This is a sample exam comprised primarily of exam questions from previous years. This
sample exam has not been gauged by 203 instructors for length, so it may be slightly longer
or shorter than the actual nal.
Sample Exam 1
EECS 203
Winter 2015
Instructions. You have 1.5 hours to complete this exam. You may use any information you
have written on a 8.511 sheet of paper. You may not use any other sources of information,
including electronic devices, textbooks, o
Sample Exam 3 EECS 203 Winter 2007
You have two hours to complete this exam. You may use any information you have written on three 8.5" 11" sheets of paper, but no other information. Leave at least one seat between yourself and other students.
Prob
EECS 203 HW 1
10. Let p, q, and r be the propositions p: You get an A on the final exam. q: You do every exercise in this book. r: You get an A in this class. Write these propositions using p, q, and r and logical connectives. a) You get an A in this
EECS 203: Discrete Mathematics
Fall 2015
Discussion 1 Notes
1. Exercise 1.1.16
Determine whether these biconditionals are true or false.
a) 2 + 2 = 4 if and only if 1 + 1 = 2
b) 1 + 1 = 2 if and only if 2 + 3 = 4
Solution:
a) 2 + 2 = 4 if and only if 1 +
Exam 1 Sample Solutions
(Part A Multiple Choice Questions)
EECS 203
Fall 2015
Name (Print):
uniqname (Print):
General Instructions
You have 90 minutes to complete the two parts of this exam. The exam is worth 100 points,
so you should work at a pace of mo
EECS 203 Supplemental Study Material for Section 5.5
5.5.12. How many different combinations of pennies, nickels, dimes, quarters, and half dollars can a piggy bank contain if it has 20 coins in it? Solution: There are 5 things to choose from, repetitions
EECS 203 HOMEWORK 2 Section1.2 Problem 10 (E): Show that each of these conditional statements is a tautology by using truth tables a) [~p(pq)]q b) [(pq) (qr)] (pr) c) [(p(pq)]q d) [(pq)(p r)(qr)] r
Section 1.2 Problem 16 (E): Show that pq and (pq)
EECS 203: Discrete Mathematics
Fall 2014
Homework 12 Solutions
EECS203 Sta
November 25, 2014
1. Section 10.5 Exercise 14
Solution:
Notice that this problem essentially asks to determine whether the corresponding graph
abstraction with is Eulerian. Because
EECS 203: Discrete Mathematics
Fall 2014
Homework 9 Solutions
EECS203 Sta
November 6, 2014
1. Section 7.4 Exercise 12
Solution:
a) Rolling n times means that for the rst (n  1) tries, it must not be a six (with
5
1
probability ), and then the last try ha
September 26, 2015
Section 1.8 Problem 22
Show that if x is nonzero real number, then x2+1/x2 2

Let x be a nonzero real number.
1 2
1
x =x 2+ 2 2
Then
x
x
( )
x 2+
x 2+

1
2 0
2
x
1
2
x2
Hence x is a nonzero real number.
Section 1.8 Problem 30
Becaus
EECS 203: Discrete Mathematics
Fall 2014
HW7 Solutions
EECS203 Sta
1. Exercise 6.4.10 Using the Binomial Theorem we have
1
x+
x
100
100
100
(x)100j
j
=
j=0
100
1
x
j
100 1002j
x
j
=
j=0
To nd the coecient of xk for k Z, rst note that for any xk that appea
L12: Permutations &
Combinations
EECS 203: Discrete Mathematics
A permutation lock!
Winter 2017
UM EECS 203 L12
1
Permutations & Combinations
n! = n(n1)(n2)321
Permutations
P(n,k) = Number of ways to choose k things (order
counts!) out of n things
Exam 1
(Part A Multiple Choice Questions)
EECS 203
Winter 2017
Name (Print):
uniqname (Print):
General Instructions
You have 90 minutes to complete the two parts of this exam. The exam is worth 100 points,
so you should work at a pace of more than a point
EECS 203: Discrete Mathematics
Winter 2017
Discussion 1 Notes
1. Exercise 1.1.15
Let p, q, and r be the propositions
p : Grizzly bears have been seen in the area.
q : Hiking is safe on the trail.
r : Berries are ripe along the trail.
Write these propos
Exam 1
(Part A Multiple Choice Questions)
EECS 203
Fall 2016
Name (Print):
uniqname (Print):
General Instructions
You have 90 minutes to complete the two parts of this exam. The exam is worth 100 points,
so you should work at a pace of more than a point p
Mathematical Induction
EECS 203: Discrete Mathematics
Lecture 9
Winter 2017
UM EECS 203 L9
1
Climbing the Ladder
We want to show that n1 P(n) is true.
Think of the positive integers as a ladder.
1, 2, 3, 4, 5, 6, . . .
You can reach the bottom of the l