Lema 19.2.6.

Misalkan \(B\) suatu aljabar Boolean dan \(a, b \in B\text{.}\) Pernyataan-pernyataan berikut ekuivalen.
  1. \(a \preceq b\text{.}\)
  2. \(a \wedge b' = O\text{.}\)
  3. \(a' \vee b = I\text{.}\)

Bukti.
(1) \(\Rightarrow\) (2). Jika \(a \preceq b\text{,}\) maka \(a \vee b = b\text{.}\) Oleh karena itu,
\begin{align*} a \wedge b' & = a \wedge (a \vee b)'\\ & = a \wedge ( a' \wedge b')\\ & = ( a \wedge a') \wedge b'\\ & = O \wedge b'\\ & = O\text{.} \end{align*}
(2) \(\Rightarrow\) (3). Jika \(a \wedge b' = O\text{,}\) maka \(a' \vee b = (a \wedge b')' = O' = I\text{.}\)
(3) \(\Rightarrow\) (1). Jika \(a' \vee b = I\text{,}\) maka
\begin{align*} a & = a \wedge (a' \vee b)\\ & = (a \wedge a') \vee (a \wedge b)\\ & = O \vee (a \wedge b)\\ & = a \wedge b\text{.} \end{align*}
Jadi, \(a \preceq b\text{.}\)
Dalam Konteks