IST4_lec10

IST4_lec10 - IST 4 Information and Logic IST 4: Planned...

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IST 4 Information and Logic
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1 M1 31 6 2 6 5 26 19 M2 5 4 12 4 5 3 M2 28 3 2 21 2 1 14 M1 7 fri thr wed tue mon IST 4: Planned Schedule – Spring 2008 x= hw#x out x= hw#x due = today T Mx= MQx out Mx= MQx due mid terms- good luck! T
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Old/Odd Machines that Compute MQ 2 Deadline Friday, 5/16/2008, 10pm Time measuring devices clocks, watches, mechanisms, history, etc. .. Weaving devices: looms, mechanisms, history, etc. .. Calculating machines: Examples are just examples, they can be ignored. ..
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So Far
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So Far The logical sense = number sense = 3 number system = syntax to extend our number sense Boolean algebra = syntax to extend our logical sense Proofs constitute an important language understand, read, write and speak
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Examples of Boolean Algebras •0-1 ±( two valued ) Boolean algebra OR / AND • Arithmetic Boolean algebras ( Boolean integers ) lcm / gcd •A lgebra±of
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Algebra of Subsets: Operations A B + is union of sets: is intersection of sets: S A A Complement
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Is the Algebra of Subsets Boolean? A B C YES A B C
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Is the Algebra of Subsets Boolean? A B C YES A B C
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Is the Algebra of Subsets Boolean? A B C YES A B C
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Examples of Boolean Algebras •0-1 ±( two valued ) Boolean algebra OR / AND • Arithmetic Boolean algebras ( Boolean integers ) lcm / gcd •A lgebra±of ±
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An Amazing Theorem Representation Theorem (Stone 1936): Every finite Boolean algebra is isomorphic to a Boolean algebra of subsets of some finite set S. Algebra 1 Algebra 2 elements elements operations/elements operations/elements Marshall Stone 1903-1989
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Algebra of Subsets 1 is the set S 0 is the empty set + is union of sets is intersection of sets Elements: all possible subsets of S (there are 2 subsets) |s| Example: Elements: S = Can be represented as: such that with and
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Algebra of Subsets 1 is the set S 0 is the empty set + is union of sets is intersection of sets Elements: all possible subsets of S (there are 2 subsets) |s| Example: Elements: S = Can be represented as: such that ?? ??
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Arithmetic Boolean Algebra The set of divisors of a Boolean integer n The operations: lcm and gcd The special elements: 1 and n Example: Elements: 1 2 3 6 Can be represented as: such that ?? ??
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Arithmetic Boolean Algebras Isomorphic to Algebra of Subsets The set of divisors of a Boolean integer: {1,2,3,6} The operations: lcm and gcd Isomorphic to: 1 2 3 6 Can be represented as: such that Can be represented as: such that 1-1 mapping is performed with x and y
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Arithmetic / Subsets 1 2 3 6 1 2 3 6 lcm 1 2 2 6 6 2 3 6 3 6 3 6 6 6 6 6
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Arithmetic / Subsets gcd 1 2 3 6 1 2 3 6 1 1 1 1 2 2 3 3 6 3 2 1 1 1 1 1
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Binary vectors - Subsets 00 10 01 11 00 10 01 11 11 01 01 01 10 10 10 00 00 00 00 00 00 00 00 00
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Elements: all possible subsets of S (there are 2 subsets) |s| Example: S = Elements: As subset can be represented by a binary vector of length |S| where a 0 represents that a point is not in the set
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IST4_lec10 - IST 4 Information and Logic IST 4: Planned...

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