Problem 14 - Discrete Mathematics with Applications

Question

Rewrite each sentence symbolically, where \(\mathrm{P}(x) : x\) is a 16 -bit machine, \(\mathrm{Q}(x) : x\) uses the ASCII" character set, and the UD \(=\) set of all computers. We can find a 16 -bit computer that does not use the ASCII character set.

Step-by-Step Solution

Answer

The short answer is: \(\displaystyle ∃x ( P( x ) ∧ ¬Q( x ) )\).

Step 1: Identify the predicates and negations

In the given sentence, we are talking about a 16-bit computer. This can be represented as P(x). The computer does not use the ASCII character set, which means we need to use the negation of Q(x), represented as ¬Q(x).

Step 2: Apply the existential quantifier

Now, we are asked to represent that there exists a 16-bit computer that does not use the ASCII character set. To do this, we will use the existential quantifier, ∃.

Step 3: Combine the predicates and quantifier

Now that we have identified the predicates and the quantifier, we can combine them into a single symbolic expression. The final symbolic representation for the given statement will be: \(\displaystyle ∃x ( P( x ) ∧ ¬Q( x ) )\) This expression states that there exists an x (a computer) such that x is a 16-bit machine (P(x)) and x does not use the ASCII character set (¬Q(x)).

Key Concepts

Predicates

In symbolic logic, predicates play a crucial role in forming statements about objects. A predicate is essentially a property or a characteristic that an object may possess. It can be thought of as a function that returns either true or false when applied to an object.
For instance, in the original exercise, we have two predicates:

• \( P(x) \): The property that \( x \) is a 16-bit machine


• \( Q(x) \): The property that \( x \) uses the ASCII character set

These predicates allow us to express complex statements involving objects (like computers) mathematically. Understanding predicates forms the basis for symbolic logic because they help decompose complicated thoughts into manageable pieces.

Existential Quantifier

The existential quantifier \( \exists \) is an essential symbol in symbolic logic, representing existence. When you see \( \exists x \), it is read as "there exists an \( x \)."
In the exercise, we want to assert that some computer is a 16-bit machine and does not use the ASCII character set.
So this leads to the statement \( \exists x(P(x) \land eg Q(x)) \).

• The quantifier \( \exists \) tells us that the statement does not need to hold for all computers, just for at least one.
  
• It provides a framework for making claims about the existence of objects with specific properties.

It is vital to differentiate this from the universal quantifier \( \forall \), which would suggest that every object in the scope has the stated properties.

Negation

Negation in symbolic logic is used to express the opposite of a given statement or predicate. It is denoted by the symbol \( eg \).
For example, when we say \( eg Q(x) \), it states that \( x \) does not have the property \( Q \), meaning that \( x \) does not use the ASCII character set. Understanding negations is key to manipulating and interpreting logical expressions.
Negation plays an important role in forming complex logical statements:

• It helps to determine what conditions do not hold true.


• When combined with quantifiers and predicates, it allows us to precisely convey specific circumstances within logical formulas.

It may seem trivial initially, but negation is powerful, especially in proofs and reasoning.

Symbolic Expressions

Symbolic expressions in logic are used to represent complex statements in a compact and unambiguous form. These expressions use logical operators, predicates, and quantifiers to capture the essence of statements.
In the provided exercise, the symbolic expression \( \exists x(P(x) \land eg Q(x)) \) effectively conveys the required meaning:

• The expression combines the existential quantifier with predicates to succinctly state the presence of a particular computer type.


• The \( \land \) symbol represents logical 'and', which implies that both conditions need to be satisfied by the computer in question.

Mastering the interpretation and construction of symbolic expressions is crucial in logic as they are the building blocks for reasoning, helping articulate complex statements clearly and effectively.