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)