Problem 13
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 is a computer that is a \(16-\) bit machine and uses the ASCII character set as well.
Step-by-Step Solution
Verified Answer
The symbolic representation of the given sentence is \(\exists c \in \text{UD}, P(c) \wedge Q(c)\), where \(P(c)\) represents a 16-bit machine, and \(Q(c)\) represents using the ASCII character set.
1Step 1: Define the predicates
\(P(x): x\) is a 16-bit machine.
\(Q(x): x\) uses the ASCII character set.
UD: Set of all computers.
2Step 2: Analyze the given sentence
The sentence states that there is at least one computer (let's call it c) that is a 16-bit machine and uses the ASCII character set.
3Step 3: Express the sentence using predicates and logical operators
To express "is a 16-bit machine and uses the ASCII character set," we can use the logical AND operator (∧) to combine the predicates P(x) and Q(x). And to express "there exists," we use the existential quantifier (∃).
Therefore, the symbolic representation of the sentence is:
\(\exists c \in \text{UD}, P(c) \wedge Q(c)\)
Key Concepts
Predicates in LogicLogical OperatorsExistential Quantifiers
Predicates in Logic
Predicates in logic are fundamental components used to formulate statements or propositions about objects. A predicate is essentially a function that takes an argument and returns a true or false value. For example, in the context of the given exercise, the predicates are:
When using predicates, we transition from specific instances to general forms using symbols. This allows us to represent complex statements logically and to determine their truth-value precisely. Understanding predicates is essential because they serve as building blocks for more complex logical statements, enabling more abstract forms of reasoning.
- \( P(x) \): \( x \) is a 16-bit machine.
- \( Q(x) \): \( x \) uses the ASCII character set.
When using predicates, we transition from specific instances to general forms using symbols. This allows us to represent complex statements logically and to determine their truth-value precisely. Understanding predicates is essential because they serve as building blocks for more complex logical statements, enabling more abstract forms of reasoning.
Logical Operators
Logical operators play a crucial role in connecting predicates to form complex logical statements. In symbolic logic, common logical operators include AND (\(\wedge\)), OR (\(\vee\)), NOT (\(eg\)), and IMPLIES (\(\rightarrow\)). These operators help in structuring sentences into more meaningful propositions.
For instance, in our exercise, the logical conjunction \(\wedge\) (AND) is used. This operator asserts that both conditions connected by it must be true for the entire expression to be true. Specifically, the statement \( P(c) \wedge Q(c) \) implies that "\( c \) is a 16-bit machine AND \( c \) uses the ASCII character set."
Mastery of logical operators allows you to analyze and build more detailed and precise logical expressions. It helps in deducing conclusions from complex statements and enhances clear communication of logic-based ideas.
For instance, in our exercise, the logical conjunction \(\wedge\) (AND) is used. This operator asserts that both conditions connected by it must be true for the entire expression to be true. Specifically, the statement \( P(c) \wedge Q(c) \) implies that "\( c \) is a 16-bit machine AND \( c \) uses the ASCII character set."
Mastery of logical operators allows you to analyze and build more detailed and precise logical expressions. It helps in deducing conclusions from complex statements and enhances clear communication of logic-based ideas.
Existential Quantifiers
Existential quantifiers are symbols used in logic to express that there is at least one element in a domain that satisfies a given predicate. The symbol \(\exists\) represents the phrase "there exists." In the context of our exercise, the statement \(\exists c \in \text{UD}, P(c) \wedge Q(c)\) indicates that there is at least one computer \( c \) that is both a 16-bit machine and uses the ASCII character set.
The power of existential quantifiers lies in their ability to express statements about existence without identifying any particular element that satisfies the condition. This abstraction is handy in various fields, such as math, computer science, and philosophy.
Using existential quantifiers effectively requires understanding their scope and how they interact with other logical components, such as predicates and logical operators. As you become proficient with existential quantifiers, you'll gain the ability to express complex ideas more succinctly and powerfully.
The power of existential quantifiers lies in their ability to express statements about existence without identifying any particular element that satisfies the condition. This abstraction is handy in various fields, such as math, computer science, and philosophy.
Using existential quantifiers effectively requires understanding their scope and how they interact with other logical components, such as predicates and logical operators. As you become proficient with existential quantifiers, you'll gain the ability to express complex ideas more succinctly and powerfully.
Other exercises in this chapter
Problem 13
Give the simplest possible conclusion in each argument. Assume each premise is true. $$\begin{aligned} &p \leftrightarrow q\\\ &\sim p \vee r\\\ &\sim r \end{al
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Prove each directly. The square of every integer of the form \(4 k+1\) is also of the same form, where \(k\) is an arbitrary integer.
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Let \(x, y,\) and \(z\) be any real numbers. Represent each sentence symbolically, where \(p: xx)]$$
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Verify each, where \(f\) denotes a contradiction. $$(p \vee q) \rightarrow r \equiv(p \rightarrow r) \wedge(q \rightarrow r)$$
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