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# 0_Version2 - 91.304 Foundations of Computer Science Chapter...

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1 91.304 Foundations of Computer Science Chapter 0 Lecture Notes Prof. David Martin and Prof. Giam Pecelli (with modifications by Prof. Karen Daniels) This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by- sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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2 About These Notes ± Designed to be used with Sipser’s Introduction to the Theory of Computation ± Available through “Lecture notes” link on course web page ± Note that examples & various other things are not included here
3 Basic Objects and Notation ± N = { 1, 2, 3, 4, . .. } ² Some texts include 0; Sipser doesn't ± Z = { . .., -2, -1, 0, 1, 2, . .. } ² Z + = N ± R = set of all real numbers ± Q = set of all rational (quotient) numbers ± PosEven = { 2n | n N } ² { x | predicate(x) } ² Tiny constraints are sometimes added to the left of |, as in ± PosEven = { n N : n % 2 = 0 } ² Sometimes : is used instead of |

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4 Scope of Intentional Notation ± Variables inside { x | pred(x) } are local ± Think of the specification as a mathematical program ² We will see many programming languages this term: DFAs, NFAs, Regex, PDAs, TMs, C++, . .. ² Mathematical notation is a type of precise specifier – i.e., a programming language ² Turns out it is far more powerful than our ordinary programming languages – we'll prove this later
5 Scope of Intentional Notation ± { 2n | n N } = { n N : n % 2 = 0 } ± These are two different programs that produce the same set different vars

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6 Set Operations ± –Un ion–D is junct ion–Or ² { 0, 3, 6, 9 } { 0, 2, 4, 6, 8 } = ? ± – Intersection – Conjunction – And ² { 0, 3, 6, 9 } { 0, 2, 4, 6, 8 } = ?
7 Set Operations ± Complement PosOdd = PosEven c ... depending ² For a set A, Universe is implicit. Be careful!!! ± Set difference PosOdd = N - PosEven = { 1, 3, 5, . ..} ² For sets A & B, A – B = { x | x A and x B } = A B c (see HW#1, Problem 1)

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8 More Set Operations ± Cardinality If A is a set, then is not very precise, but oh well. We’ll improve upon this later when we start counting infinities
9 More on Sets ± The empty set: = {} ± The number of {} matters: ² { } { { }, } ± Elements of a set ² { 5 } { 1, {2,3}, , { 5 }, 2 } ² { 3 } { 1, {2,3}, , { 5 }, 2 } ² { 1, {2,3}, , { 5 }, 2 } ² Is always a member of a set even if it is not explicitly listed? ± Subset ² { 1, { 5 } } { 1, {2,3}, , { 5 }, 2 } ² ∅⊆ { 1, {2,3}, , { 5 }, 2 } ² {2,3} * { 1, {2,3}, , { 5 }, 2 }

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10 More Set Operations ± Cartesian Product (aka Cross Product) If A and B are sets, then ² A × B = { (a,b) | a A and b B } ± Note that the × operator "preserves structure" by wrapping parentheses and commas around its arguments ± { 1,3 } × { c, d, f } = ? ± If |A| = n and |B| = m, then |A × B| = ?
11 More Set Operations ± Generalizing to more sets:

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12 More Set Operations ± Power set P (A) = 2 A = { x | x A } ± Important equivalence ² x A means the same as x P (A) ± Examples: ² P ({1,2}) = { , {1}, {2}, {1,2} } ² P ( ) = ?
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## This note was uploaded on 02/13/2012 for the course CS 91.304 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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0_Version2 - 91.304 Foundations of Computer Science Chapter...

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