IST4_lec10

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

This preview shows pages 1–21. Sign up to view the full content.

IST 4 Information and Logic

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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. ..

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
So Far
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Examples of Boolean Algebras •0-1 ±( two valued ) Boolean algebra OR / AND • Arithmetic Boolean algebras ( Boolean integers ) lcm / gcd •A lgebra±of
Algebra of Subsets: Operations A B + is union of sets: is intersection of sets: S A A Complement

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Is the Algebra of Subsets Boolean? A B C YES A B C
Is the Algebra of Subsets Boolean? A B C YES A B C

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Is the Algebra of Subsets Boolean? A B C YES A B C
Examples of Boolean Algebras •0-1 ±( two valued ) Boolean algebra OR / AND • Arithmetic Boolean algebras ( Boolean integers ) lcm / gcd •A lgebra±of ±

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 ?? ??
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 ?? ??

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 68

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

This preview shows document pages 1 - 21. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online