Topic 9
Ch 5.7, 7.1, 7.3, 7.4 in Grimaldi
Relations
 Properties of Relations
 Partial Orders: Hasse Diagrams
 Equivalence Relations and Partitions
 Growth of Functions and Big O notation
Relations
Any subset of AB is called a relation from A to
B

A
Topic 4
Ch 4.3  4.5 in Grimaldi
Application of concepts:
 divisibility and primes
Number Theory
Division:
If a and b are integers where a 0, then a
divides b (denoted ab) if and only if there
is an integer c such that b = ac. When a
divides b, then
a
Equivalence Relations
Recall:
 A relation R on set S is an equivalence
relation if R is reflexive, symmetric, and
transitive
Equivalence Class:
 Let R be an equivalence relation on set A:
 The set of all elements that are related
to an element aA is ca
Topic 4
Ch 4.3  4.5 in Grimaldi
Application of concepts:
 divisibility and primes
1
Number Theory
Division:
If a and b are integers where a 0, then a
divides b (denoted ab) if and only if there
is an integer c such that b = ac. When a
divides b, then
Name:
Student ID:
1. Let p,q,r,s denote the following
p: I finish writing my computer program before lunch
q: I play tennis
r: it is sunny
s: it is soggy
Write the following in logic form
a) I will play tennis if and only if it is sunny.
b) Finishing the
Macm 101 Problem Set 1
1. With n a positive integer, evaluate the sum
C(n,0) + 2C(n,1) + 22C(n,2) + . + 2kC(n,k) + . + 2nC(n,n)
2. Show that if n and r are positive integers, then C(n+1, r) = (n+1)C(n,r1)/
Name:
Student ID:
1. Establishthevalidity(orinvalidity)ofthefollowingstatements.Ifvalid,showusingthelaws
oflogic.Otherwisegiveacounterexample:
a. p(pq)
qr.
r
b. p(qr)
ps
tq
s.
rt
c. pq
q
p(qr)
r
2. Lettheuniversebemadeupofallshapes
r(x)=xisarectangle
s(
Topic 3
Ch 2.4  2.5 in Grimaldi
Proofs and Quantifiers
 use of Quantifiers
 Proofs of theorems
Next week:
 Application of concepts: divisibility and primes
Quantifiers
 open statements
Recall: the following is not a proposition (statement)
x is an in
Macm 101 Problem Set 3
questions in Grimaldi
2.3 Q 112
2.4 Q 19, 1122
2.5 Q 110, 1224
More difficult Questions:
1. Give the truth value of the following in the universe of integers x, y. Briefly explain
your answer:
a) x y ( x = 1/y )
b) x y ( y2 x < 100
Topic 2
Ch 2.1  2.3 in Grimaldi
Fundamentals of Logic
 Logical Connectives and Truth Tables
 Laws of Logic (equivalence)
 Rules of inference (implication)
 Methods of proofs
Logical Connectives
 Basic connectives
Statements (propositions) are declar
Macm 101 Midterm Equation Sheet DO NOT WRITE ON THIS PAPER!
Counting Equations:
Formula
Permutation
nr
With repetition
n!
n r!
No repetition
n!
n1 !n2 !nr !
No repetition, repeated objects
Combination
n
n!
=
r
r!nr!
No repetition
With repetition
n+r
Name:
Student ID:
1. Let p,q,r,s denote the following
p: I finish writing my computer program before lunch
q: I play tennis
r: it is sunny
s: it is soggy
Write the following in logic form
a) I will play tennis if and only if it is sunny.
b) Finishing the
Problem Set 6
Practice questions in Grimaldi
3.2 Q 115, 1720
3.3 Q 13, 510
3.4 Q 15, 710
More difficult Problems
1. Let A, B, and C be sets in some universe U. Prove or disprove the following using
a membership table
Topic 1
Grimaldi: Ch 1.1  1.4
Principles of Counting
 sum and product rules
Permutations
 with/without repetition
Combinations
 the Binomial and Multinomial theorems
 with repetition
Principles of Counting
Counting (enumeration) is a way to computing
Topic 6
Not in text
Pseudo code and intro to algorithms:
Pseudocode
 algorithm
An algorithm is a finite set of precise instructions for
solving a problem or carrying out a task
An algorithm involving a discrete structure is usually
described using pseudo