cs 454 theory of computation sonoma state university, fall 2011 instructor: b. (ravi) ravikumar...
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CS 454 Theory of Computation
Sonoma State University, Fall 2011
Instructor: B. (Ravi) Ravikumar
Office: 116 I Darwin HallOriginal slides by Vahid and Givargis, Mani Srivastava and othersExtensive editing by Ravikumar
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Lecture 2
Goals:• chapter 0 (45 minutes)• Quiz 1 (10 minutes)• Chapter 1 (rest of the class)
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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.
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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}, }
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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?
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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.
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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)}
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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.
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Some special relations• equivalence relation• partial-order relation
Graphs:• Definition• Example
Strings and languages• Boolean logic• quantifiers
Summary of mathematical terms
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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.
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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.
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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?
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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?
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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?
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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.
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• 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
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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.
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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: 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.
2
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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.
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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 (counter-example) 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.