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## 2-element Boolean algebra

Name: $\mathbb B_2=\langle\{0,1\},\vee,\wedge,',0,1\rangle$

Elements: 0,1

0=0

1=1

##### Unary operations
 $x$ $x'$ 0 1 1 0
##### Binary operations

Join = or =

$\vee$01
001
111

Meet = and = multiplication =

$\wedge$01
000
101
##### Derived operations

Symmetric difference: $x\oplus y=(x\vee y)\wedge-(x\wedge y)$

Implication: $x\to y=-x\vee y$

Bi-implication: $x\leftrightarrow y=(x\to y)\wedge(y\to x)$

##### Properties
Simple Yes Yes
##### Notes

This algebra generates the variety of all Boolean algebras.

Every Boolean algebra is a subdirect product of $\mathbb B_2$.