Homework 7

1

• Homework 7
• Average 70 Median 75
• http//www.cs.virginia.edu/cmt5n/cs202/hw7/
• Homework 8
• Average 74 Median 79
• http//www.cs.virginia.edu/cmt5n/cs202/hw8/
• 36 was commonly missed
• The solution key presents a rather tedious,
  straightforward application of the counting
  techniques we discussed in class.
• There is a more clever way to solve this problem
  but it takes a bit of creative insight beyond
  what we discussed in class. I did not present
  this more clever solution in the key because I
  want you to see how it can be solved without this
  insight.
• Emmanuels last OH is this Thursday, April 24th

2
Review for Final Exam (Part 1)

• Propositional Logic
• Know the rules and connectives
• Be able to translate statements into
  propositional logic
• Be able to produce logical equivalences
• Solve Logic Puzzles (Whos Lying?,
  Knights/Knaves, etc.)
• Predicate Logic
• Know the rules, quantifies, and format for
  predicate logic
• Be able to translate statements into predicate
  logic
• Know how to form common statements (such as at
  least two, etc)

3

• Methods of Proof
• Know how to apply the different proof techniques
• When can you use a counter-example, when do you
  need UG?
• Be able to prove given statements by applying
  the proper method
• Know how to spot invalid use of a proof
  technique
• Sets
• Know the definition of commonly used sets N,
  Z, Z, R, Q, etc.
• Be able to work with set operations ?, ?, ?,
  ?, ?, P(S), etc.
• Have facility with set specifiers , ?, ?, ?,
  ?, S, etc.
• Know set terminology empty, disjoint, etc.
• Understand the bit string representation of a
  subset of a given set
• Review things that gave you trouble on TEST1

4

• Functions
• KNOW THE DEFINITION OF MAP AND FUNCTION!
• Be able to tell whether a function is 1-1 or
  onto and show it
• Understand function operations such as
  composition and inverse
• What is a 1-1 correspondence? How does it
  relate to counting?
• The Integers and Division
• Know the definitions a b, even, odd,
  rational, irrational, prime, composite, mod,
  relatively prime, pairwise relatively prime
• Review the theorems in this section regarding
  divisibility, mod, primes, fundamental theorem of
  arithmetic, division algorithm, etc.
• Be able to find the prime factorization, gcd,
  lcm, mod, etc.
• Prove results concerning divisibility, rational,
  irrational, prime, composite, mod, etc.

5

• Proof Strategy
• The main goal is to be able to prove theorems.
  Strategies presented include working backwards
  from the desired result, using existing proofs,
  looking for counter-examples, making conjectures
• Sequences and Summations
• Review sequences, summations, summation
  notation, cardinality
• Know the definition of countability,
  uncountability, etc.
• How do sequences relate to countability?
• Be able to determine whether a set is countable
  or uncountable.
• Prove results concerning countability/uncountabil
  ity.
• Understand the technique of diagonalization and
  how it is used to prove that a set is
  uncountable.
• Other methods of proving a set is uncountable?

6
Ex Show that the set of irrational numbers is
uncountable.
We can not use the technique of diagonalization
here because we dont have a rigorous
representation for irrational numbers.
Proof Assume, to the contrary, that the set of
irrational numbers is countable. Use I to
represent the set of irrational numbers.
We know that R Q ? I. We also know that Q is
countable.
Hence R is countable as the union of two
countable sets we proved it.
But we know that R is uncountable we proved this
as well.
This is a contradiction a set cant be both
countable and uncountable, so the set of
irrational numbers must be uncountable, after
all.?
Take specific note here that diagonalization is
not the only means by which to prove a set is
countable. Also note that just because we may
not be able to diagonalize a set doesnt mean it
isnt uncountable.
7

• Mathematical Induction
• Know the principle of mathematical induction
• When can we use mathematical induction
• Be able to prove results using mathematical
  induction
• Recursive Definitions and Structural Induction
• KNOW HOW A RECURSIVE SET DEFINITION WORKS!
• Be able to determine the elements of a
  recursively defined set
• Understand the technique of structural induction
  and how to use it to prove results about
  recursively defined sets and sequences.
• Review things that gave you trouble on TEST2
• Pay specific attention to understanding
  countability/uncountability
• Also the workings of recursive set definition
  gave a lot of trouble.