Question

Interpret the remaining three of Boole's equations \(x \bar{y} z=0\), \(x \bar{y} \bar{z}=0, \bar{x} y z=0\) from the case in the text, where \(x\) stands for clean beasts, \(y\) for beasts that divide the hoof, and \(z\) for beasts that chew the cud.

Short answer

Interpret the remaining Bool's equations in the context of clean beasts, beasts that divide the hoof, and beasts that chew the cud. Equation 1: \(x\bar{y}z=0\) represents clean beasts that don't divide the hoof but chew the cud. Since all clean beasts must both divide the hoof and chew the cud, there are no such beasts, so the equation equals 0. Equation 2: \(x\bar{y}\bar{z}=0\) represents clean beasts that neither divide the hoof nor chew the cud. This goes against the definition of clean beasts, so there are no such beasts and the equation equals 0. Equation 3: \(\bar{x}yz=0\) represents unclean beasts that divide the hoof and chew the cud. However, any beast that divides the hoof and chews the cud is considered clean. So, there are no such unclean beasts with these characteristics, and the equation equals 0.

Step-by-step solution

Equation 1: x.bar(y)z = 0

Here, \(x\bar{y}z=0\) represents clean beasts that don't divide the hoof but chew the cud. As all clean beasts must both divide the hoof and chew the cud (based on the description), we can say there are no such beasts, so the equation equals 0.

Equation 2: x.bar(y).bar(z) = 0

In this equation, \(x\bar{y}\bar{z}=0\), we have clean beasts that neither divide the hoof nor chew the cud. Since this goes against the definition of clean beasts, it means that there are no such beasts, and the equation equals 0.

Equation 3: bar(x).y.z = 0

Lastly, for the equation \(\bar{x}yz=0\), it represents unclean beasts that divide the hoof and chew the cud. However, that's not possible because any beast that divides the hoof and chews the cud is considered clean. So, there are no such beasts that are unclean yet also have these characteristics. Thus, the equation equals 0.

Key concepts

George Boole

George Boole was a pioneering mathematician and logician who laid the foundational work for what we now know as Boolean algebra.
He was born in the 19th century and his groundbreaking work allowed for the development of digital logic used in computers today.
Boole wanted to figure out how logical statements could be expressed in mathematical terms. Some key contributions by Boole include:

• Establishing a systematic way to solve logical problems using a form of algebra known as Boolean algebra.
• Introducing variables to represent logical statements, where such variables can only have values true or false, equivalent to 1 and 0 respectively.
• Formulating laws and principles (such as the law of identity and law of excluded middle) which are foundational for computer science.

Therefore, Boole's work is fundamental in digital circuitry and computer programming.
His approach allows us to "compute" logical propositions, such as the ones found in our exercise, through mathematical equations.

logical equations

Logical equations, like those used by George Boole, are expressions that relate logical variables to each other.
These are mathematical representations of logic statements used to identify truths within a logical framework.
The equations in our exercise demonstrate this concept clearly. To break it down:

• The equation \(x \bar{y} z = 0\) implies the condition in which clean beasts do not divide the hoof but chew the cud.
• Similarly, \(x \bar{y} \bar{z} = 0\) shows clean beasts that neither divide the hoof nor chew the cud.
• Finally, \(\bar{x} y z = 0\) explores unclean beasts that have both characteristics of dividing the hoof and chewing the cud.

Each of these logical equations is designed to represent different combinations of conditions.
By setting these conditions equal to zero, we state that such combinations or situations are impossible under given logical rules.
Logical equations allow us to formalize and solve logical problems using algebraic methods.

mathematical logic

Mathematical logic is the study and application of logic within mathematics.
It uses mathematical symbols and methods to examine the fundamental nature of truth, inference, and logical structures.
This branch of logic includes theories of computation, proof, and set theories which are crucial for understanding complex logical scenarios. In the context of the exercise, mathematical logic helps:

• Translate verbal logical statements into mathematical equations for clearer interpretation.
• Identify and resolve logical inconsistencies using Boolean algebra.
• Simplify logical reasoning by using algebraic methods to determine the validity of propositions.

The equations demonstrate how mathematical logic can be used to deduce truths about theoretical environments.
It helps to precisely articulate problems concerning combinations and comparisons.
Ultimately, mathematical logic supports a wide range of applications including computer science, where it aids the development and functioning of algorithms and programming.