MACM 101 Final Exam
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Student Number:
Signature:
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/100
There are a total of 100 points possible on this exam.
1. (6 points) Consider the statement \If x is a perfect square
Homework 8
MACM 101
April 2, 2014
Date due: April 09, 2014.
PART A: Practice questions. Some of the questions in this part will be covered
in the tutorials. You are strongly encouraged to work on these problems. You
are not required to hand in the solutio
10-2
Discrete Mathematics Theorems and Proofs
Previous Lecture
Axioms and theorems
Rules of inference for quantified statements
Direct proofs
Theorems
Introduction and Proofs
Discrete Mathematics
Andrei Bulatov
Discrete Mathematics Theorems and Proofs
10-
MIDTERM
When: Wednesday, October 17, regular time 11:30
Relations
Introduction
Where: Here
Topics covered:
- Propositional logic
- Predicates and quantifiers
- Sets and set operations
Midterm will include problems on propositional logic
Discrete Mathemati
14-2
Discrete Mathematics Equivalences
Previous Lecture
Sets and logic
Laws of set theory
Cartesian products of two and more sets
Cardinality and other properties of Cartesian products
Relations
Introduction
Discrete Mathematics
Andrei Bulatov
13-3
Discre
Discrete Mathematics Orders and Equivalences
15-2
Previous Lecture
Cartesian product, the cardinality of Cartesian product
Binary relations, higher arity relations
Describing binary relations
list of pairs
matrix of relation
graph of relation
Properties
I
Discrete Mathematics Mathematical Induction
21-2
Principle of Mathematical Induction
Climbing an infinite ladder
Can we reach every step of it, if
Mathematical Induction
Introduction
For all k, standing on the
rung k we can step on
the rung k + 1
We can r
Discrete Mathematics Combinations
25-4
Combinations (cntd)
An r-combination of elements of a set is an unordered selection
of r elements from the set. Thus, an r-combination is simply a
subset of the set with r elements.
The number of r-combinations of a
Discrete Mathematics - Permutations
24-4
Permutations
Given a collection of n distinct objects, any (linear) arrangement
of these objects is called a permutation of the collection.
Introduction
Permutations
A permutation of size r (0 r n) is any (linear)
Discrete Mathematics - Sets
11-2
What is a Set
A set is an unordered collection of objects
This is not a rigorous definition!
Every `conventional definition reduces the defined concept to a
wider, more general, concept.
Sets
Introduction
For example,
`A c
12-2
Discrete Mathematics Operations on Sets
Previous Lecture
Sets and elements
Subsets, proper subsets, empty sets
Universe
Cardinality
Power set
Operations
Introduction on Sets
Discrete Mathematics
Andrei Bulatov
12-3
Discrete Mathematics Operations on
Name: - Student Id: Group:
Midterm 1 (MACM101-D1)
February 3, 2014.
Test duration: 50 minutes
There are eight questions in this test. Answer questions worth 60 points.
1. (10 points) Consider selecting 3 objects from the set A = cfw_1, 2, 3, 4, 5, 6.
(a)
Pigeonhole Principle
includes
solu0ons to PartB ques0ons
Homework #7
Date due: April 2, 2014
The pigeonhole principle (PHP)
If m pigeons occupy n pigeonholes, and m > n, then at least
one pigeonhole has two or
Name: - Student Id: Group:
Midterm 2 (MACM101-D1)
March 10, 2014.
Test duration: 50 minutes
Total: 35 points
The bonus question is worth 5 points.
1. (8 points) Our universe is Z+ .
(a) Prove that the largest number you cannot write as the sum of 5 or 9
Solutions for Homework 5
MACM 101
Due: Mar 12, 2014
PART B:
1. Prove by mathematical induction
1) Basis Case: when n = 1, we have
! !
!
!
Since ! ! ! ! , and together with ! , applying Modus
Ponens we know ! is valid.
2) Inductive steps:
Assume S(k) is
2-2
Discrete Mathematics Propositional Logic
What is Logic?
Computer science is continuation of logic by other means
Georg Gottlob
Propositional Logic
Introduction
Contrariwise, continued Tweedledee, if it was so, it might be; and if it
were so, it would
Discrete Mathematics Predicates and Quantifiers II
8-2
Previous Lecture
Predicates
Assigning values, universe, truth values
Predicates and Quantifiers II
Introduction
Discrete Mathematics
Andrei Bulatov
7-3
Discrete Mathematics Predicates and Quantifiers
Discrete Mathematics Rules of Inference
5-2
Previous Lecture
Logically equivalent statements
Statements and are equivalent iff is a tautology
Main logic equivalences
Rules of
Introduction Inference
double negation
DeMorgans laws
commutative, associative,
Discrete Mathematics Pigeonhole Principle
27-2
Pigeonhole Principle
If m pigeons occupy n pigeonholes and m > n, then at least one
pigeonhole has two or more pigeons roosting in it.
Pigeonhole Principle
Introduction
Discrete Mathematics
Andrei Bulatov
Dis