IST 4: Planned Schedule – Spring 2009
The logical sense = number sense = 3
number system = syntax to extend our number sense
Boolean algebra = syntax to extend our logical sense
Leibniz never held an academic position…
Boole was never a student at a university…
only a professor…
The Babylonian never did homework sets...
So Far
Proofs are FUN…
especially when they are short… and correct...
Current state
Associativity Theorem
Theorem 3:
Proof: (ideas)
The architecture:
DeMorgan Theorem
Theorem 4:
Proof:
Need to prove: 
Idea: need to validate A2 
Need to prove:
Proof:
Q: Simple proof of
DeMorgan without
Associativity??
Extra Credit: 10 pt
No collabo ration
Current state
Proof F low
Examples of Boolean Algebras
0-1 ( two valued )
Boolean algebra
OR / AND
Arithmetic Boolean
algebras (Boolean integers)
lcm / gcd
Algebra of subsets
union / intersection
Boole’s Original Motivation
Two- valued Boolean Algebra
 |
| 0 iff both x and y are 0 |
1 iff both x and y are 1 |
Is a Boolean algebra
Non-Binary Boolean Algebras
Elements:
The set of divisors of a Boolean integer
{1,2,3,5,6,10,15,30}
The operations: lcm and gcd
The special elements: 1 and 30
Algebra of Subsets
A set is a collection of points in a domain
Universal set: S is the set of all points
For example: S is the set of all books |
 |
Subsets:
A = red books
B = blue books
C = green books
D = German books
E = English books
F = French books
G = good books
Euler Diagram
Venn Diagram
John Venn
1834 - 1923
Leonhard Euler
1707-1783
graphical re presentation
of all possible subsets
