Boolean algebra

Boolean algebra was created by George Boole and is a very powerful tool to describe and design logic circuits. In Boolean algebra the elements can only assume the binary values: $0$ or $1$.
Boolean algebra is based on the following set of axioms:

$$\begin{array}{rlrl}& 1a.0\cdot 0=0& & 1b.1+1=1\\ & 2a.1\cdot 1=1& & 2b.0+0=0\\ & 3a.0\cdot 1=1\cdot 0=0& & 3b.1+0=0+1=1\\ & 4a.\text{If }x=0\text{, then }\overline{x}=1& & 4b.\text{If }x=1\text{, then }\overline{x}=0\\ & 5a.x\cdot 0=0& & 5b.x+1=1\\ & 6a.x\cdot 1=x& & 6b.x+0=x\\ & 7a.x\cdot x=x& & 7b.x+x=x\\ & 8a.x\cdot \overline{x}=0& & 8b.x+\overline{x}=1\\ & 9a.\overline{\overline{x}}=x& & \end{array}$$

We also have some properties to deal with multiple variables:

$$\begin{array}{rlrlrlr}& 10a.x\cdot y=y\cdot x& & 10b.x+y=y+x& & & \text{Commutative}\\ & 11a.x\cdot (y\cdot z)=(x\cdot y)\cdot z& & 11b.x+(y+z)=(x+y)+z& & & \text{Associative}\\ & 12a.x\cdot (y+z)=x\cdot y+x\cdot z& & 12b.x+y\cdot z=(x+y)\cdot (x+z)& & & \text{Distributive}\\ & 13a.x+x\cdot y=x& & 13b.x\cdot (x+y)=x& & & \text{Absorption}\\ & 14a.x\cdot y+x\cdot \overline{y}=x& & 14b.(x+y)\cdot (x+\overline{y})=x& & & \text{Combining}\\ & 15a.\overline{x\cdot y}=\overline{x}+\overline{y}& & 15b.\overline{x+y}=\overline{x}\cdot \overline{y}& & & \text{DeMorgan}\\ & 16a.x+\overline{x}\cdot y=x+y& & 16b.x\cdot (\overline{x}+y)=x\cdot y& & & \\ & 17a.x\cdot y+y\cdot z+\overline{x}\cdot z=x\cdot y+\overline{x}\cdot z& & 17b.(x+y)\cdot (y+z)\cdot (\overline{x}+z)=(x+y)\cdot (\overline{x}+z)& & & \text{Consensus}\end{array}$$