# Definition:Two (Boolean Algebra)

*This page is about Two in the context of Boolean Algebra. For other uses, see Two.*

## Definition

Denote with $\top$ the canonical tautology.

Denote with $\bot$ the canonical contradiction.

Define $\mathbf 2 := \set {\bot, \top}$, read **two**.

When endowed with the logical operations $\lor$, $\land$ and $\neg$, $\mathbf 2$ becomes a Boolean algebra.

These operations have the following Cayley tables:

- $\begin{array}{c|cc} \lor & \bot & \top \\ \hline \bot & \bot & \top \\ \top & \top & \top \end{array} \qquad \begin{array}{c|cc} \land & \bot & \top \\ \hline \bot & \bot & \bot \\ \top & \bot & \top \end{array} \qquad \begin{array}{c|cc} & \bot & \top \\ \hline \neg & \top & \bot \end{array}$

That $\mathbf 2$ thus becomes a Boolean algebra is shown on Two is Boolean Algebra.

## Also known as

Some sources use $0$ and $1$ for $\bot$ and $\top$, respectively.

The three operations $\vee$, $\wedge$ and $\neg$ still have the same Cayley tables, so that this is a matter of convention only.

This article is complete as far as it goes, but it could do with expansion.The intimate connection with classical PropCalc needs to be expanded uponYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

Work In ProgressNeeds to move, technically, to Definition:Two (Boolean Lattice)You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

Define $\preceq$ to be the ordered set determined by putting $\bot \preceq \top$.

When endowed with the logical operations $\lor$, $\land$ and $\neg$, $\mathbf 2$ becomes a Boolean lattice.

## Also see

- Definition:Two (Category), the category $\mathbf 2$ becomes when viewed as an order category.

## Sources

- 2008: Paul Halmos and Steven Givant:
*Introduction to Boolean Algebras*... (previous) ... (next): $\S 2$