Fundamental mathematical tools

• Boolean Algebra and Truth Tables are essential important tools to express logical relationships .
• Truth Tables : Provides a listing of every possible combination of values of binary inputs to a digital circuit and the corresponding outputs.
• Boolean Algebra
  • Elements : true or false ( 1, 0)
  • Operations: a OR b; a AND b, NOT a
  
  

Single-Variable Theorems

Switching-algebra theorems with one variable

Example

• 一般使用perfect induction（全部列举）来证明

Two- and Three-Variable Theorems

• ATTENTION :
  • (X + Y) · (X + Z) = X + Y · Z

  (X + Y)(X + Z) = X(X + Z) + Y(X + Z) (using the first distributive law)
  = XX + XZ + YX + YZ (using the first distributive law)
  = X + XZ + YX + YZ (using the idempotent law)
  = X·1 + XZ + YX + YZ (using the operation with 1 law)
  = X(1 + Z + Y) + YZ (using the first distributive law)
  = X·1 + YZ (using the operation with 1 law)
  = X + YZ (using the operation with 1 law)

• XY’ + Y = X + Y

XY’ + Y = XY’ + 0 + Y(1+X)
= XY’ + YY’ + YY+YX
= (X + Y)·Y’+ (X + Y)·Y
= (X + Y)·(Y’+Y)
= X + Y

• XY + X’Z + YZ = XY + X’Z

XY + X’Z + YZ = XY + X’Z
= XY + X’Z + (X + X’)YZ
= XY + X’Z + XYZ + X’YZ
= XY + XYZ + X’Z + X’YZ
= XY(1 + Z) + X’Z(1 + Y)
= XY·1 + X’Z·1
= XY + X’Z

Examples
W = [M + N’P + (R + ST)’][M + N’P + R + ST]
X = M + N’P
Y = R + ST
W = (X + Y’)(X + Y)
= XX + XY + Y’X + Y’Y
= X+ XY + XY’ + 0
= X + X(Y + Y’) = X + X·1 = X
W = M + N’P
变量可以分组的时候考虑替换掉

Important Words

Boolean Algebra ——布尔代数
Truth Tables ——真值表
Perfect induction ——完备归纳法
Identities——自等律
Null elements——0-1律
Idempotency——同一律
Involution ——还原律
Complements——互补律
Commutativity——交换律
Associativity——结合律
Distributivity——分配律
Covering——吸收律
Combining——组合律
Consensus——一致律