Name: $\mathbb B_2=\langle\{0,1\},\vee,0,\wedge,1,'\rangle$
Elements: 0,1
0=0
1=1
Complement = negation = $1-x$ =
| $x$ | 0 | 1 |
|---|---|---|
| $x'$ | 1 | 0 |
Alternative notation: $-x=\overline x=x^-=\neg x$
Join = or = truncated addition = $\min\{x+y,1\}$ =
| $\vee$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 1 |
| 1 | 1 | 1 |
Meet = and = multiplication =
| $\wedge$ | 0 | 1 |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 0 | 1 |
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 |
| Simple | Yes |
|---|---|
| Subdirectly irreducible | Yes |
This algebra generates the variety of all Boolean algebras.
Every Boolean algebra is a subdirect product of $\mathbb B_2$.
none
4-element Boolean algebra $\mathbb B_2^2$
1-element Boolean algebra $\mathbb B_1$
4-element Boolean algebra $\mathbb B_2^2$
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