Problem 16 - 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. There exists a computer that is neither a 16 -bit machine nor uses the ASCIl character set.

Step-by-Step Solution

Answer

The short answer for the given question is: \(\exists x (\lnot\mathrm{P}(x) \land \lnot\mathrm{Q}(x))\)

Step 1: Identify the given predicates

We are given \(\mathrm{P}(x): x\) is a 16-bit machine and \(\mathrm{Q}(x) : x\) uses the ASCII character set.

Step 2: Identify the given UD

The UD (universal domain) is the set of all computers.

Step 3: Use the existential quantifier

To express "there exists a computer," we use the existential quantifier ∃. In this case, the variable x represents a computer.

Step 4: Express the negation of both conditions

To express that the computer is neither a 16-bit machine nor uses the ASCII character set, we will negate both predicates and use the conjunction ∧ to connect them. So, we have ¬P(x) ∧ ¬Q(x).

Step 5: Combine the existential quantifier with the negated conditions

To complete the symbolic representation of the statement, we combine the existential quantifier and the negated conditions: ∃x (¬P(x) ∧ ¬Q(x)) This represents "There exists a computer (x) that is neither a 16-bit machine nor uses the ASCII character set."

Key Concepts

Predicates

In symbolic logic, a predicate is a statement involving a variable that can be true or false depending on the value of the variable. It typically describes a property or a relation between entities. For example, in our exercise, we have two predicates:

• \( P(x) \) represents the statement that "\( x \) is a 16-bit machine".
• \( Q(x) \) signifies the proposition that "\( x \) uses the ASCII character set".

These predicates are functions of \( x \), which can take on various values within a given domain.
Predicates are crucial in constructing logical expressions as they allow more complex conditions to be expressed about variables.

Existential Quantifier

The existential quantifier is a fundamental operator in logic used to indicate that a predicate holds true for at least one element in a specified domain.
It is symbolized by an inverted \( \forall \), which is \( \exists \). In the context of the problem, "there exists a computer..." is expressed as \( \exists x \).
This tells us that there is at least one computer within our specified domain, which is the set of all computers, for which the subsequent logical conditions are true.

Negation

Negation is a logical operation that converts a statement to its opposite in terms of truth value.
When negating a predicate, the operation is symbolized by \( eg \), which is placed before the predicate to indicate "not."
In this exercise, we negate both predicates because the statement involves a computer that "is neither a 16-bit machine nor uses the ASCII character set."
So we write it as \( eg P(x) \) and \( eg Q(x) \), indicating that the characteristics specified by \( P(x) \) and \( Q(x) \) are not true for the variable \( x \).

Conjunction

Conjunction is a binary operation used to combine two logical statements, which will be true if and only if both constituent statements are true.
The symbol for conjunction is \( \wedge \), representing the logical "and." In our exercise, the conjunction \( eg P(x) \wedge eg Q(x) \) signifies that both negated conditions are met for a particular computer \( x \).
Therefore, \( x \) must neither be a 16-bit machine nor use the ASCII character set, both conditions must hold simultaneously.

Universal Domain

The universal domain, often abbreviated as UD, represents the set of all potential subjects or entities under discussion in a logical problem.
In this exercise, the universal domain is "the set of all computers," meaning that any variable \( x \) will refer to some computer.
This context helps us understand what kinds of objects our predicates and quantified statements (like those using \( \exists \)) are concerning.
The specification of the universal domain ensures that the logical expressions are evaluated in a consistent and relevant context.