Question

Recall that in the definition of a boolean algebra, we did not require that \(T, \perp,\) and each \(\neg x\) be specified; we merely said they must exist. So, it is natural to ask whether there might be several clements that could equally well be chosen as \(T\) or \(\perp\) or, for some clement \(x\) of the boolean algebra, several different possible choices for \(\neg x\), Show that in a complemented laftice: (a) There is only one possible choice of elements \(T\) and \(\perp\) satisfying the definition of a complemented lattice, (Hint: Suppose there were two possible choices for \(T\). say, \(T_{1}\) and \(T_{2}\). Evaluate \(T_{1} \wedge T_{2}\) in two different ways. ) (b) For each element \(x\) of a complemented, distributive lattice, there is only one possible choice for \(\neg x\) that satisfies the definition of \(\neg x\). (Hint: Suppose there were two choices, say, \(\neg x_{1}\) and \(\neg x_{2},\) for \(\neg x\). Find two ways to evaluate \(\neg x_{1} \wedge x \vee \neg x_{2}\).)

Short answer

In a complemented lattice, \( T \) and \( \perp \) are unique, and each element has a unique complement.

Step-by-step solution

Define the Problem

We need to show that in a complemented lattice, there can only be one choice for the elements \( T \) and \( \perp \), and for each element \( x \), there is only one possible choice for \( eg x \).

Uniqueness of T

Assume there are two elements, \( T_1 \) and \( T_2 \), both satisfying the identity of \( T \) in the lattice. So, according to the properties, for any element \( a \): \( a \wedge T_1 = a \) and \( a \wedge T_2 = a \). Evaluate \( T_1 \wedge T_2 \). Since \( T_2 \) acts as a universal bound, \( T_1 \wedge T_2 = T_1 \). Also \( T_1 \) acts as a universal bound, so \( T_1 \wedge T_2 = T_2 \). Therefore, \( T_1 = T_2 \).

Uniqueness of ⊥

Similarly, assume there are two elements, \( \perp_1 \) and \( \perp_2 \), both satisfying the identity of \( \perp \) in the lattice. For any element \( a \), we have \( a \vee \perp_1 = a \) and \( a \vee \perp_2 = a \). Evaluate \( \perp_1 \vee \perp_2 \). Since \( \perp_2 \) acts as a null element, \( \perp_1 \vee \perp_2 = \perp_2 \). Also \( \perp_1 \) acts as a null element, so \( \perp_1 \vee \perp_2 = \perp_1 \). Thus, \( \perp_1 = \perp_2 \).

Uniqueness of ¬x

For an element \( x \), assume there are two complements, \( eg x_1 \) and \( eg x_2 \). By definition of lattice complement, \( eg x_1 \vee x = T \) and \( eg x_2 \wedge x = \perp \). Evaluate the expression \( (eg x_1 \wedge x) \vee (eg x_2 \wedge x) \): this simplifies to \( \perp \). Now use distributivity: \( (eg x_1 \wedge x) \vee (eg x_2 \wedge x) = (eg x_1 \vee eg x_2) \wedge x \). For \( eg x_1 \vee eg x_2 \) to satisfy \( (eg x_1 \vee eg x_2) \wedge x = \perp \) for all \( x \), \( eg x_1 = eg x_2 \) must hold.

Key concepts

Complemented Lattice

In the realm of Boolean Algebra, a complemented lattice is a specific structure that allows us to work with notions of complementarity in logic. This lattice is made up of elements that exhibit certain operations, primarily meet (\(\wedge\)) and join (\(\vee\)), and it satisfies properties akin to logical complementation.

One of the crucial aspects of a complemented lattice is the existence of two special elements: \(T\) (the top element) and \(\perp\) (the bottom element). In such a lattice, \(T\) and \(\perp\) serve as "universal bounds," meaning for any element \(a\) in the lattice, combining \(a\) with \(T\) using meet yields \(a\) itself, and combining \(a\) with \(\perp\) using join yields \(a\) again.

Importantly, the complemented lattice demands that these elements are unique. To understand why, consider the assumption of having two distinct top elements, \(T_1\) and \(T_2\). By the definition of a top element, \(a \wedge T_1 = a\) and \(a \wedge T_2 = a\) hold for any element \(a\). Checking the meet operation between \(T_1\) and \(T_2\), both express the identity, leading to a conclusion that \(T_1 = T_2\). Likewise, a similar logic applies to \(\perp\).

This uniqueness assures us that operations within a complemented lattice are consistent and predictable, preserving logical integrity.

Distributive Lattice

A distributive lattice is another important structure within the study of Boolean Algebra. In a distributive lattice, the two operations, meet (\(\wedge\)) and join (\(\vee\)), adhere to the distributive laws. This means the structure maintains logical relationships similar to arithmetic: for instance, \(a \wedge (b \vee c) = (a \wedge b) \vee (a \wedge c)\).

The distributive property plays a significant role when we consider complements within such a lattice. In a distributive lattice, every element \(x\) has a unique complement \(eg x\). The complement satisfies the conditions \(x \vee eg x = T\) and \(x \wedge eg x = \perp\).

To see why this uniqueness is important, imagine two potential complements for an element \(x\), named \(eg x_1\) and \(eg x_2\). The distributive law helps us evaluate the compound expression \((eg x_1 \wedge x) \vee (eg x_2 \wedge x)\), which ultimately simplifies to \(\perp\), confirming that \(eg x_1 = eg x_2\) to satisfy the requirements across all elements and operations. Thus, distributive lattices ensure that all logical operations remain coherent and reliable.

Uniqueness of Elements

The notion of uniqueness is a cornerstone in a complemented and distributive lattice. It ensures that logical operations yield consistent and clear results. Let's break down why this is essential for Boolean algebra:

• Uniqueness of Top and Bottom Elements: In any premium lattice, the top element \(T\) and the bottom element \(\perp\) are unique. They serve as references for all other operations and linkages within the lattice. Their uniqueness establishes a framework within which all elements can be logically organized.
• Uniqueness of Complements: For each element \(x\) in a complemented lattice, the complement \(eg x\) must be unique. This ensures that the expression for negation possesses definite outcomes, maintaining logical symmetry and allowing easy reversibility.

Uniqueness brings predictability and prevents ambiguous interpretations, which are critical for complex logical computations. It paves the way for reliably constructing and deconstructing logical expressions, forming the basis for theoretical and practical applications of Boolean algebra.