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. In particular: The intimate connection with classical PropCalc needs to be expanded upon You 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 Progress In particular: Needs 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): $\S 2$