# Definition:Two (Boolean Algebra)

*This page is about the Boolean algebra $\mathbf 2$. For other uses, see Definition:Two.*

## Contents

## Definition

Denote with $\top$ the canonical tautology.

Denote with $\bot$ the canonical contradiction.

Define $\mathbf 2 := \left\{{\bot, \top}\right\}$, 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.

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

- 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): $\S 2$