[Digital Circuit] algebraic structures에 사용되는 Basic Common Postulates (cf. 분배법칙 증명)

1. Closure 

- 정의) A set S is closed with respect to a binary operator if, for every pair of elements of S, the binary operator specifies a rule for obtaining a unique elements of S

 

2. Associate law(결합법칙)

 - (x*y)*z=x*(y*z) for all x,y,z∈S

 

3. Commutative law(교환법칙)

 - x*y=y*x for all x,y∈S

 

4. Identity elements(항등원)

 - for all x∈S, e*x=x*e=x 

 

5. Inverse(역원)

 - for all x∈S , exists y∈S, such that x*y=e 

   이때의 y를 x의 inverse(역원)이라 한다. 

 

6. Distributive law(분배법칙) 

 - (a)  x(y+z) = xy+xz

  (b) x+yz=(x+y)(x+z)

- (b) 식이 와닿지 않아서 증명해보았다. 

(x+y)(x+z) = xx+xz+yx+yz

              = x+xz+yx+yz

              = x(1+z) + yx+yz

              = x+yx+yz            <1+z = 1>

              = x(1+y)+yz

              = x+yz