George Boole

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 Idempotent Complement Commutativity Associativity Distributivity Absorption DeMorgan

Proof using Truth Tables X 0 1 XY XY

X X 0 1

X Y XY

X Y X Y

Finger Exercise 1

Finger Exercise 2

AB House Alarm Problem A 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 closed A = 0 the door is open B = 1 someone is on the pressure pad B = 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

AB distributive complement redundancy distributive complement House Alarm Problem (ctd)