Download Chapter 0 - Ravikumar - Sonoma State University

CS 454 Theory of Computation
Sonoma State University, Fall 2011
Instructor: B. (Ravi) Ravikumar
Office: 116 I Darwin Hall
Original slides by Vahid and Givargis, Mani Srivastava and others
Extensive editing by Ravikumar
Lecture 2
Goals:
• chapter 0
• Quiz 1
• Chapter 1
(45 minutes)
(10 minutes)
(rest of the class)
Chapter 0
• Sets
• Set operations and set relations
• union, intersection, complement
• member, subset, equality
• Venn diagrams
Problem:
Show that (A U B) C = (A C)U (B C)
Proof:
Let x be in LHS set. Then, x is in both A U B and is in
C. i.e., x is A and C, or x is in B and C. I.e., x is in A C
or in B  C. This means,
x is in (A  C)U (B  C). Converse is similar.
Sequences and tuples
A sequence is a list of objects where order is important.
Thus, <1, 2, 4, 10> is a sequence that is different than <2, 1,
4, 10>.
A finite sequence of length k is called a k-tuple. Thus, the
above sequence is a 4-tuple.
• Power-set of a set A is the set of all its subsets.
Ex: A = {1, 2} P(A) = {{1}, {2}, {1,2}, f}
Cartesian product
A X B = {<i,j> | i is in A and j is in B }
Example: A = {a, b, c}, B = { 1, 2}
What is A X B?
Cartesian product
A X B = {<i,j> | i is in A and j is in B }
Example: A = {a, b, c}, B = { 1, 2}
What is A X B?
A x B = { <a, 1>, <a, 2>, <b, 1>, <b, 2>, <c, 1>, <c, 2>}
A X A X … X A is often denoted by Ak.
Relations
A relation on a set A is a subset of A X A.
Example 1: A = { 1, 2, …, 10}, relation R is defined as: (a,
b) is in R if a – b = 3.
R = { (4, 1), (5, 2), (6, 3), (7, 4), (8, 5), (9, 6), (10, 7)}
Example 2: Children’s game of Rock-Paper-Scissors.
Relation could be “beats”.
B = { (scissors, paper), (paper, rock), (rock, scissors)}
Functions
A function (from A to B) is a relation R in which for every
i, there is a unique j such that <i,j> is in R.
• onto: for any y in B, there is at least one x in A such
that <x, y> is in R.
• one-one: for any y in B, there is at most one x in A
such that <x, y> is in R.
• bijective: for any y in B, there is exactly one x in A
such that <x, y> is in R.
Some special relations
• equivalence relation
• partial-order relation
Graphs:
• Definition
• Example
Strings and languages
• Boolean logic
• quantifiers
Summary of mathematical terms
Definitions, theorems and proofs
• Definitions: A definition is the way to describe an object
in a way that its characteristics are completely captured in
the description.
• Assertions: Mathematical statement expresses some
property of a set of defined objects. Assertions may or may
not be true.
• Proof: is a convincing logical argument that a statement
is true. The proofs are required to follow rigid rules and
are not allowed any room for uncertainties or ambiguities.
• Theorem: is a proven assertion of some importance.
Rules for carrying out proof
Usually assertions are compound statements that are connected
using Boolean connectives and quantifiers. You can use theorems of
Boolean logic in the proof.
Direct proof:
• For example, if the assertion is P or Q, you can show it as follows:
Suppose P is not true. Then, Q must be true.
• Similarly, to show P  Q, you assume P is true. From this, show Q
is true. To show that P <-> Q, you should show P  Q and Q  P.
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Is this a definition? Is it an assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
It is an assertion. Is it a true assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Yes, it is a true assertion.
How to prove this assertion?
Example:
Every positive integer is either a prime, or is a product
of two integers both of which are strictly smaller than
itself.
Yes, it is a true assertion.
How to prove this assertion?
This requires knowing the definition of a prime number.
Prime number: A number x is prime if it has exactly two
divisors, namely 1 and x.
• Most of the theorems assert properties of a collection of
objects.
• If the collection is finite, usually it is easy:
• you can show it for every member one by one.
• Need for proof really arises when the assertion is of the form :
“Every object in a set X has some property Y.” where X is an
infinite set.
Now you can’t prove it one by one!
• Proof techniques: (not an exhaustive list!)
• proof by construction
• proof by contradiction
• proof by induction
Proof by construction:
Definition: A k-regular graph is a graph in which every node has
degree k.
Theorem: For each even number n greater than 2, there exists a
3-regular graph with n nodes.
Although the construction below shows the claim for n = 14, it
can be readily generalized.
Proof by contradiction
To show P  Q, assume Q is not true, then show that P is false,
which contradicts the hypothesis that P is true.
Theorem: 2 is not rational, i.e., it can’t be written as a ratio of two
integers.
A proof by contradiction is presented in page 22.
Alternate proof: (This is not a proof by contradiction.) This proof is
based on two lemmas.
Lemma 1: A positive integer n is a perfect square if and only if it has
an odd number of divisors.
Lemma 2: If a positive odd integer n has d divisors, then 2n has 2d
divisors.
Proof by induction
Example 1: Formula for mortgage calculation.
Page 24 of text.
Example 2: Show that the set of all binary strings of length n can be
arranged in a way that every adjacent string differs in exactly one bit
position, and further the first and the last string also differ in exactly
one position.
For n = 2, one such is 00, 01, 11, 10.
Assertion: Every integer is a sum of squares of two integers.
This is not true. To disprove it, it is enough to find one integer (counterexample) that can’t be written as sum of two squares.
Consider 3. Suppose x2 + y2 = 3 for some integers x and y. This means,
x2 is either 0, 1 or 2. (Why not any other number?)
Case 1: x2 is 0. Thus y2 is 3. But there is no integer y such that y2 = 3.
(How can we show this?)
Case 2: x2 is 1. Thus y2 is 2. But from previous slide, we know that
there is no integer y such that y2 = 2.
Case 3: x2 is 2. But from previous slide, we know that there is no
integer x such that x2 = 2. End of proof.