Postulates (1.2)

Note: $ means “there exists”, " means “for all”.

1. Digital Abstraction: " x Î Â = {0,1},

–  x=0 if x ¹ 1

–  x=1 if x ¹ 0

2. Existence of complements: " x ÎÂ,

$ x’ Î  (called the complement of x) such that

–  x + x’ = 1

3. Truth tables for •, +, ‘

– 0+0=0, 0+1=1+0=1, 1+1=1

– 0•0=0, 0•1=1•0=0, 1•1=1

– 0’=1, 1’=0

4. Closure: " x, y ÎÂ

– x+y ÎÂ _ – x•y Î Â_ – x

5. Existence of identity elements

– " x ÎÂ _ , $ 0 ÎÂ _ such that x+0 = 0+x = x.

– " x ÎÂ _ , $ 1 ÎÂ _ such that x•1 = 1•x = x.

6. Commutative laws: " x, y ÎÂ,

– x+y = y+x

– x•y = y•x

7. Distributive laws: " x, y, z ÎÂ,

» x•(y+z) = x•y+x•z

» x+(y•z) = (x+y)•(x+z)