George Boole 1815-1864
Set Theory
More Set Theory – Union (“OR”)X is the set of men, Y is the set of women
X + Y = ?
Y + X = ?
More Set Theory – Intersection (“AND”)
X is the set of living things, Y is the set of cats
X.Y = ?
Y.X = ?
Even More Set Theory – Fundamental Equation
X is the set of cats
X.X = ?
X.X = X
What numbers work?
Set Theory – Summary
X + Y = Y + X
X . X = X
X . Y = Y . X
X + Y = Y + X
X . X = X
X . Y = Y . X
Sets Numbers
Involution
Identity
IdempotentComplement
CommutativityAssociativityDistributivityDistributivity AbsorptionDeMorgan
Proof using Truth Tables
𝑋+𝑋=1
X 01
𝑋+𝑌=𝑋 .𝑌X Y 0 00 11 01 1
X Y 0 00 11 01 1
X
𝑋 .𝑋=0X 01
X Y
𝑋+𝑋 .𝑌=𝑋X Y 0 00 11 01 1
X Y
(𝑋 .𝑌 )=𝑋+𝑌
X Y
Finger Exercise 1
𝐵 . ( 𝐴+𝐵 )=𝐵 . 𝐴
Finger Exercise 2
( 𝐴+𝐵 ) . ( 𝐴+𝐵 )=𝐴
A B0 00 11 01 1
House Alarm ProblemA house alarm has a sensor on the door (A) and a pressure pad inside the house (B).
The logical values of the sensors are:
A = 1 the door is closedA = 0 the door is openB = 1 someone is on the pressure padB = 0 no-one is on the pressure pad The alarm sounds when the door is open, or if someone is on the pressure pad, or both
A B0 0 10 1 11 0 01 1 1
distributivecomplementredundancydistributivecomplement
House Alarm Problem (ctd)
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