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Chapter 1: Problem 2

Which of these are propositions? What are the truth values of those that are propositions? a) Do not pass go. b) What time is it? c) There are no black flies in Maine. d) \(4+x=5\) . e) The moon is made of green cheese. f) \(2^{n} \geq 100\) .

Short Answer

Expert verified
a) No, b) No, c) Yes, True/False, d) Yes, True/False, e) Yes, False, f) Yes, True/False.

Step by step solution

01

Define what a proposition is

A proposition is a declarative statement that is either true or false.
02

Evaluate statement a)

a) 'Do not pass go.' - This is not a declarative statement; it is an imperative one. Therefore, it is not a proposition.
03

Evaluate statement b)

b) 'What time is it?' - This is a question, not a declarative statement. Therefore, it is not a proposition.
04

Evaluate statement c)

c) 'There are no black flies in Maine.' - This is a declarative statement. If it is true, its truth value is True; if it is false, its truth value is False. It is a proposition.
05

Evaluate statement d)

d) \(4+x=5\) - This is a mathematical equation which is a declarative statement. It is true if \(x=1\) and false otherwise. It is a proposition.
06

Evaluate statement e)

e) 'The moon is made of green cheese.' - This is a declarative statement. Since this is not true, its truth value is False. It is a proposition.
07

Evaluate statement f)

f) \(2^{n} \geq 100\) - This is a mathematical inequality statement. It is true if \(n \geq 7\) and false otherwise. It is a proposition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Declarative Statements
Declarative statements are fundamental in understanding propositions in logic. They are sentences that state a fact, opinion, or a condition that can be either true or false. For example, the statement 'There are no black flies in Maine' is declarative. It makes a clear claim about the world that can be verified. Non-declarative sentences, such as questions ('What time is it?') or commands ('Do not pass go.'), do not qualify as propositions because they do not have a truth value. That鈥檚 why identifying declarative statements is the first step in logical analysis.
Truth Values
In logic, each proposition has a truth value: True or False. Determining the truth value is crucial for logical analysis. For example, the statement 'The moon is made of green cheese.' is false because we know that the moon is not composed of green cheese. On the other hand, mathematical statements like the equation \(4+x=5\) can be true or false depending on the value of \(x\). When \(x=1\), the equation is true; otherwise, it is false. Providing clarity on the truth value helps solidify understanding of propositions.
Logical Analysis
Logical analysis involves evaluating whether statements are propositions and determining their truth values. Start by identifying if the statement is declarative. Then, assess its truth value. For instance, the statement 'There are no black flies in Maine.' is declarative. Its truth value can be true or false based on real-world conditions. Similarly, the statement \(2^{n} \geq 100\) is logical to evaluate: if \(n\geq 7\), the statement is true; otherwise, it is false. Logical analysis helps develop critical thinking and reasoning skills by breaking down complex information into manageable parts.
Mathematical Statements
Mathematical statements are specialized propositions involving mathematical expressions. They follow the same rules as other propositions in determining truth values. For instance, \(4+x=5\) is a mathematical statement. We can verify its truth value by solving for \(x=1\). Another example is the inequality \(2^{n}\geq 100\). It becomes true when \(n\geq 7\). Understanding mathematical statements and their validity is essential for applying logical concepts in mathematics and solving problems efficiently.

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Most popular questions from this chapter

Translate these statements into English, where \(R(x)\) is "x is a rabbit" and \(H(x)\) is " \(x\) hops" and the domain consists of all animals. $$ \begin{array}{ll}{\text { a) } \forall x(R(x) \rightarrow H(x))} & {\text { b) } \forall x(R(x) \wedge H(x))} \\ {\text { c) } \quad \exists x(R(x) \rightarrow H(x))} & {\text { d) } \exists x(R(x) \wedge H(x))}\end{array} $$

Determine the truth value of each of these statements if the domain consists of all integers. $$ \begin{array}{ll}{\text { a) } \forall n(n+1>n)} & {\text { b) } \exists n(2 n=3 n)} \\ {\text { c) } \exists n(n=-n)} & {\text { d) } \forall n(3 n \leq 4 n)}\end{array} $$

Construct a truth table for each of these compound propositions. a) \(p \rightarrow(\neg q \vee r)\) b) \(\neg p \rightarrow(q \rightarrow r)\) c) \((p \rightarrow q) \vee(\neg p \rightarrow r)\) d) \((p \rightarrow q) \wedge(\neg p \rightarrow r)\) e) \((p \rightarrow q) \vee(\neg q \rightarrow r)\) f) \((\neg p \leftrightarrow \neg q) \leftrightarrow(q \leftrightarrow r)\)

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) When there is less than 30 megabytes free on the hard disk, a warning message is sent to all users. b) No directories in the file system can be opened and no files can be closed when system errors have been detected. c) The file system cannot be backed up if there is a user currently logged on. d) Video on demand can be delivered when there are at least 8 megabytes of memory available and the connection speed is at least 56 kilobits per second.

Prove that there are 100 consecutive positive integers that are not perfect squares. Is your proof constructive or non-constructive?
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