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Chapter 1: Problem 14

Let \(p, q,\) and \(r\) be the propositions \(p :\) You have the flu. \(q :\) You miss the final examination. \(r :\) You pass the course. Express each of these propositions as an English sentence. a) \(p \rightarrow q\) b) \(\neg q \leftrightarrow r\) c) \(q \rightarrow \neg r\) d) \(p \vee q \vee r\) e) \((p \rightarrow \neg r) \vee(q \rightarrow \neg r)\) f) \((p \wedge q) \vee(\neg q \wedge r)\)

Short Answer

Expert verified
a) If you have the flu, then you miss the final examination. b) You do not miss the final examination if and only if you pass the course. c) If you miss the final examination, then you do not pass the course. d) You have the flu, or you miss the final examination, or you pass the course. e) Either if you have the flu, then you do not pass the course, or if you miss the final examination, then you do not pass the course. f) Either you have the flu and you miss the final examination, or you do not miss the final examination and you pass the course.

Step by step solution

01

Understanding the Propositions

First, understand the given propositions: - Proposition 饾憹 means 'You have the flu'. - Proposition 饾憺 means 'You miss the final examination'. - Proposition 饾憻 means 'You pass the course'.
02

Convert: 饾憹 鈫 q

For 饾憹 鈫 q, this is translated as 'If you have the flu, then you miss the final examination.'
03

Convert: 卢饾憺 鈫 饾憻

For 卢饾憺 鈫 饾憻, 卢饾憺 means 'You do not miss the final examination'. So this can be translated as 'You do not miss the final examination if and only if you pass the course.'
04

Convert: 饾憺 鈫 卢饾憻

For 饾憺 鈫 卢饾憻, 卢饾憻 means 'You do not pass the course'. So this can be translated as 'If you miss the final examination, then you do not pass the course.'
05

Convert: 饾憹 鈭 饾憺 鈭 饾憻

For 饾憹 鈭 饾憺 鈭 饾憻, this is translated as 'You have the flu, or you miss the final examination, or you pass the course.'
06

Convert: (饾憹 鈫 卢饾憻) 鈭 (饾憺 鈫 卢饾憻)

For (饾憹 鈫 卢饾憻) 鈭 (饾憺 鈫 卢饾憻), 饾憹 鈫 卢饾憻 is 'If you have the flu, then you do not pass the course', and 饾憺 鈫 卢饾憻 is 'If you miss the final examination, then you do not pass the course'. So this is translated as 'Either if you have the flu, then you do not pass the course, or if you miss the final examination, then you do not pass the course.'
07

Convert: (饾憹 鈭 饾憺) 鈭 (卢饾憺 鈭 饾憻)

For (饾憹 鈭 饾憺) 鈭 (卢饾憺 鈭 饾憻), 饾憹 鈭 饾憺 is 'You have the flu and you miss the final examination', and 卢饾憺 鈭 饾憻 is 'You do not miss the final examination and you pass the course'. So this is translated as 'Either you have the flu and you miss the final examination, or you do not miss the final examination and you pass the course.'

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Key Concepts

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

Logical Connectives
In propositional logic, logical connectives are symbols or words that connect propositions to form more complex sentences. The main logical connectives used include:
  • Conjunction (鈭): means 'and'
  • Disjunction (鈭): means 'or'
  • Negation (卢): means 'not'
  • Implication (鈫): means 'if...then'
  • Biconditional (鈫): means 'if and only if'
These connectives allow us to create compound propositions and analyze logical statements in a structured manner.
Propositional Equivalences
Propositional equivalences are tools that help simplify complex logical expressions by identifying when two statements are logically the same. Some common equivalences include:
  • Double Negation Law: 卢(卢p) is equivalent to p
  • De Morgan's Laws: 卢(p 鈭 q) is equivalent to (卢p 鈭 卢q) and 卢(p 鈭 q) is equivalent to (卢p 鈭 卢q)
  • Commutative Laws: p 鈭 q is equivalent to q 鈭 p and p 鈭 q is equivalent to q 鈭 p
  • Distributive Laws: p 鈭 (q 鈭 r) is equivalent to (p 鈭 q) 鈭 (p 鈭 r)
  • Implication: p 鈫 q is equivalent to 卢p 鈭 q
Using these equivalences, we can transform and simplify logical statements in a meaningful way, ensuring clarity and correctness.
Truth Tables
Truth tables are essential tools in propositional logic, used to determine the truth value of logical expressions based on the truth values of their components. Each row of a truth table represents a possible combination of truth values for the propositions involved. For example, the truth table for a simple proposition p and its negation 卢p would be:
  • If p is True (T), then 卢p is False (F)
  • If p is False (F), then 卢p is True (T)
By filling out a truth table, you can visualize and verify the behavior of more complex logical statements.
Conjunction
Conjunction is a logical connective that corresponds to 'and' in natural language and is symbolized by 鈭. For propositions p and q, the conjunction p 鈭 q is true only when both p and q are true. For instance:
  • If p: 'You have the flu' is true
  • And q: 'You miss the final examination' is true
  • Then p 鈭 q: 'You have the flu and you miss the final examination' is true
Conversely, if either p or q is false, then p 鈭 q is false. This connective helps express combined conditions in logical formulations.
Disjunction
Disjunction is another fundamental logical connective represented by 鈭. It translates to 'or' in natural language. The disjunction p 鈭 q is true if at least one of the propositions p or q is true. For example:
  • If p: 'You have the flu' is false
  • But q: 'You miss the final examination' is true
  • Then p 鈭 q: 'You have the flu or you miss the final examination' is true
Disjunction is used to connect scenarios where meeting either one of the described conditions suffices.
Negation
Negation, represented by 卢, is a logical operation that inverts the truth value of a proposition. For a proposition p, the negation 卢p is true if p is false, and 卢p is false if p is true. For example:
  • If p: 'You have the flu' is true
  • Then 卢p: 'You do not have the flu' is false
Negation plays a crucial role in forming the complement of a given proposition and is fundamental in logical operations and proofs.
Implication
Implication, symbolized by 鈫, expresses a conditional relationship between two propositions. For p 鈫 q, the implication is true if whenever p is true, q is also true. It is only false when p is true, but q is false. For example:
  • If p: 'You have the flu' is true
  • Then q: 'You miss the final examination'
  • So p 鈫 q: 'If you have the flu, then you miss the final examination' is true
Implications are used to establish cause-and-effect relationships in logical arguments.
Biconditional
A biconditional, represented by 鈫, indicates that two propositions are logically equivalent; that is, p 鈫 q is true if both p and q are either true or false. For example:
  • If p: 'You have the flu' and q: 'You miss the final examination' are both true
  • Then p 鈫 q: 'You have the flu if and only if you miss the final examination' is true
Biconditionals are useful in asserting that two propositions are dependent on each other. They must both hold the same truth value for the biconditional to be true.

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Most popular questions from this chapter

What is wrong with this argument? Let \(S(x, y)\) be "\(x\) is shorter than \(y\) ." Given the premise \(\exists s S(s, \text { Max })\) , it follows that \(S(\text { Max, Max })\) . Then by existential generalization it follows that \(\exists x S(x, x),\) so that someone is shorter than himself.

Express each of these statements using logical operators, predicates, and quantifiers. a) Some propositions are tautologies. b) The negation of a contradiction is a tautology. c) The disjunction of two contingencies can be a tautology. d) The conjunction of two tautologies is a tautology.

Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase "It is not the case that.") a) No one has lost more than one thousand dollars playing the lottery. b) There is a student in this class who has chatted with exactly one other student. c) No student in this class has sent e-mail to exactly two other students in this class. d) Some student has solved every exercise in this book. e) No student has solved at least one exercise in every section of this book.

Determine whether \(\forall x(P(x) \leftrightarrow Q(x))\) and \(\forall x P(x) \leftrightarrow\) \(\forall x Q(x)\) are logically equivalent. Justify your answer.

Use forward reasoning to show that if \(x\) is a nonzero real number, then \(x^{2}+1 / x^{2} \geq 2 .[\text { Hint: Start with the in- }\) equality \((x-1 / x)^{2} \geq 0,\) which holds for all nonzero real numbers \(x . ]\)
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