2-element Boolean algebra
Name: B2=⟨{0,1},∨,0,∧,1,′⟩
Elements: 0,1
Constant operations
0=0
1=1
Unary operations
Complement = negation = 1−x =
| x | 0 | 1 |
|---|---|---|
| x′ | 1 | 0 |
Alternative notation: −x=¯x=x−=¬x
Binary operations
Join = or = truncated addition = min =
| \vee | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
Meet = and = multiplication =
| \wedge | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
Derived operations
Symmetric difference: x\oplus y=(x\vee y)\wedge(x\wedge y)' = (x\wedge y')\vee(y\wedge x')
| \oplus | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 0 |
Implication: x\to y=x'\vee y
| \to | 0 | 1 |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 0 | 1 |
Bi-implication: x\leftrightarrow y=(x\to y)\wedge(y\to x)
| \leftrightarrow | 0 | 1 |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 0 | 1 |
Nand: x|y=(x\wedge y)'
| | | 0 | 1 |
|---|---|---|
| 0 | 1 | 1 |
| 1 | 1 | 0 |
Nor: x\downarrow y=(x\vee y)'
| \downarrow | 0 | 1 |
|---|---|---|
| 0 | 1 | 0 |
| 1 | 0 | 0 |
Properties
| Simple | Yes |
|---|---|
| Subdirectly irreducible | Yes |
Basic results
This algebra generates the variety of all Boolean algebras.
Every Boolean algebra is a subdirect product of \mathbb B_2.
Maximal subalgebras
none
Minimal superalgebras
4-element Boolean algebra \mathbb B_2^2
Maximal homomorphic images
1-element Boolean algebra \mathbb B_1
Minimal homomorphic preimages
4-element Boolean algebra \mathbb B_2^2
Maximal subvarieties
Minimal supervarieties
???
