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Question

# A Boolean algebra may also be defined as a partially ordered set with certain additional properties. Let $(B, \leqslant)$ be a partially ordered set. For any $x, y \in B,$ we define the least upper bound of x and y as an element z such that $x \leqslant z, y \leqslant z,$ and if there is any element $Z^{*}$

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Given:

\begin{align*} x+y&=z\text{ if z is the least upper bound of x and y} \\ x\cdot y&=z\text{ if z is the greatest lower bound of x and y} \end{align*}

$z$ is the $\textbf{least upper bound}$ of $x$ and $y$ when:

$\bullet$ $x\preceq z$

$\bullet$ $y\preceq z$

$\bullet$ if there is some element $z^*$ such that $x\preceq z^*$ and $y\preceq z^*$, then $z\preceq z^*$

$z$ is the $\textbf{greatest lower bound}$ of $x$ and $y$ when:

$\bullet$ $z\preceq x$

$\bullet$ $z\preceq y$

$\bullet$ if there is some element $z^*$ such that $z^*\preceq x$ and $z^*\preceq y$, then $z^*\preceq z$

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