A Mathematical Logician
By: Hanan Mohammed
George John Boole (1815-1864)
Objectives:
Who is this guy?
Why should we know him?
Impact on Mathematical and Logical fields
Who is George John Boole ?
Sun of shoes a seller and a lady maid
John interest in mathematics
Start going to school
Interests in education?
Eldest sibling,…so?
Who is George BooleA student & A teacher
Finance obstacles and educational
Became teacher assistant to support family
Serious mathematics learning start!
Establish his own school
Published in
Cambridge Mathematical Journal
Duncan Gregory, editor
Who is George BooleA Star Mathematician
Queens College Mathematics professor
First Mathematics professor
Science department Dean
Several calculus accomplishments
Applied algebra to solve differential equations
Who is George BooleBoolean algebra maker
Algebra of logic: Boolean algebra
Regarded Logic as aspect of Mathematics
Mathematical Analysis of Logic
Discovered analogy between algebraic
symbols and logic forms
An Investigation of the Laws of Thought (1854), on Which are Founded
the Mathematical Theories of Logic and Probabilities
Why do we care Represent Logic
as mathematical formulas
• Manipulated as normal algebraic
expressions
• Value of input and output is:
true/false
• Horned and sheep example (=x & y)
Why do we care Boolean Algebra
Rules
• P1: X = 0 or X = 1
• P2: 0 . 0 = 0
• P3: 1 + 1 = 1
• P4: 0 + 0 = 0
• P5: 1 . 1 = 1
• P6: 1 . 0 = 0 . 1 = 0
• P7: 1 + 0 = 0 + 1 = 1
Why do we care Boolean Algebra
Laws
• Idempotent Law
– X+X=X
– X X=X
• Involution Law
– 0’=1
– 1’=0
– (X’)’=X
• Complementarily Law
– X+X’=1
– X X’=0
Why do we care Boolean Algebra
Laws
• Associative Law
– (X+Y)+Z = X+(Y+Z) = X+Y+Z
– (X Y) Z = X (Y Z) = X Z Y
• Distributive Law
– X (Y+Z) = X Y+X Z
– X+(Y Z) = (X+Y) (A+Z)
• Commutative Law
– X+Y=Y+X
– X Y=X Y
Why do we care Boole's work is the
basis of mechanisms
Claude Shannon after 70 years extends
Boole's studies
Design system electromechanics through
Boolean algebra
Solved Boolean Algebra through circuits