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.

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.