Commutativity, Distributive laws, Associativity, DeMorgan's laws, Absorption (Covering), Consensus, Clutching (1.7)

Commutativity

Commutativity: " x, y ÎÂ,

– x+y = y+x

– x•y = y•x

Distributive laws

Distributivity: " x, y, z ÎÂ

» x•(y+z) = x•y + x•z

» x + (y•z) = (x+y)•(x+z)

Provable via perfect induction

Associativity

·         (x+y) + z = x + (y+z)

·         (x•y)•z = x•(y•z)

DeMorgan's laws

·         THESE ARE IMPORTANT

·         (x+y)’ = x’•y’

·         (x•y)’ = x’+y’

Absorption (Covering)

·         x + x•y = x

·         x•(x+y) = x

·         Proof: x + x•y  = x•1 + x•y = x•(1+y) = x•(y + 1)= x•1 = x

QED  (second part true by duality)

Consensus

·         xy + x’z + yz = xy + x’z

·         (x+y)•(x’+z)•(y+z) = (x+y)•(x’+z)

·         Proof: xy + x’z + yz  = xy + x’z + (x+x’)yz = xy + x’z + xyz + x’yz= (xy + xyz) + (x’z + x’zy)  = xy + x’z                                 

      QED.

Clutching

• xy+xy’=x
• (x+y)(x+y’)=x