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Chapter 1: Problem 6

Use rules of inference to show that the hypotheses 鈥淚f it does not rain or if it is not foggy, then the sailing race will be held and the lifesaving demonstration will go on,鈥� 鈥淚f the sailing race is held, then the trophy will be awarded,鈥� and 鈥淭he trophy was not awarded鈥� imply the conclusion 鈥淚t rained.鈥�

Short Answer

Expert verified
It rained because the hypotheses logically imply that it did.

Step by step solution

01

Define the Statements

Translate each part of the exercise into logical statements. Let: - \( p \): It rains - \( q \): It is foggy - \( r \): The sailing race will be held - \( s \): The lifesaving demonstration will go on - \( t \): The trophy will be awarded
02

Express the Hypotheses in Logical Form

Using the notation from Step 1, express each hypothesis as a logical statement: 1. \((eg p \lor eg q) \rightarrow (r \land s)\) 2. \(r \rightarrow t\) 3. \(eg t\)
03

Apply Modus Tollens to Hypotheses 2 and 3

Using Modus Tollens on \(r \rightarrow t\) and \(eg t\), infer \(eg r\): - If \(r\) were true, then \(t\) would be true by hypothesis 2. - Since \(t\) is false (hypothesis 3), \(r\) must be false. - Conclusion: \(eg r\).
04

Apply Modus Tollens to Hypothesis 1

Using \((eg p \lor eg q) \rightarrow (r \land s)\) and \(eg r\): - If \(r\) is false, then \((r \land s)\) is also false. - Since hypothesis 1 implies \((r \land s)\), this means \((eg p \lor eg q)\) is false. - Conclusion: \((eg(eg p \lor eg q))\).
05

Simplify the Conclusion

Simplify \(eg(eg p \lor eg q)\) using De Morgan's Laws: - \(eg(eg p \lor eg q)\) is equivalent to \(p \land q\). - Conclusion: \(p \land q\).
06

Derive the Final Conclusion

From \(p \land q\), infer that \(p\) must be true. Therefore, it rained. - Conclusion: \(p\).

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Key Concepts

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

logical statements
Logical statements are the building blocks of logical reasoning. They are sentences that are either true or false, but not both. In our exercise, we defined several logical statements based on given scenarios:
  • ewline Let \( p \): It rains
  • \( q \): It is foggy
  • \( r \): The sailing race will be held
  • \( s \): The lifesaving demonstration will go on
  • \( t \): The trophy will be awarded
By translating verbal scenarios into logical statements, we can manipulate and analyze them using formal logic rules.
modus tollens
Modus Tollens is a fundamental rule of inference in logic. It allows us to conclude the negation of a premise if an implication and its consequent are known to be true and false respectively. It follows this structure:
  • If \( A \rightarrow B \), and if \( eg B \) is true, then \( eg A \) is also true.
In our exercise, we used Modus Tollens twice:
  • From hypothesis 2 (\( r \rightarrow t \)) and hypothesis 3 (\( eg t \)), we inferred \( eg r \).
  • We then applied it again to hypothesis 1, together with \( eg r \), to infer that \( p \land q \).
Modus Tollens is powerful because it helps us deduce what must be false based on certain given truths.
De Morgan鈥檚 laws
De Morgan's Laws are essential for simplifying and transforming logical expressions. They show how to distribute negation inside and outside of parentheses involving \( \land \) and \( \lor \) operators. The laws are stated as:
  • \( eg (A \land B) \equiv eg A \lor eg B \)
  • \( eg (A \lor B) \equiv eg A \land eg B \)
In our exercise, we applied one of these laws to simplify our conclusion:
  • We had \( eg (eg p \lor eg q) \).
  • Applying De Morgan鈥檚 Law, we transformed it to \( p \land q \).
This simplification was crucial for arriving at the final conclusion that it rained.
hypotheses and conclusion
In logical reasoning, hypotheses are the starting assumptions or premised statements. Conclusions are the results deduced from these hypotheses using rules like Modus Tollens and De Morgan's Laws. In our exercise, we started with three hypotheses:
  • \( (eg p \lor eg q) \rightarrow (r \land s) \)
  • \( r \rightarrow t \)
  • \( eg t \)
From these, we deduced the conclusion \( p \): It rained. Each step involved applying logical rules systematically to reveal what must be true or false. This method ensures conclusions are logically sound and based on given information.

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Most popular questions from this chapter

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers. a) The sum of two negative integers is negative. b) The difference of two positive integers is not necessarily positive. c) The sum of the squares of two integers is greater than or equal to the square of their sum. d) The absolute value of the product of two integers is the product of their absolute values.

A statement is in prenex normal form (PNF) if and only if it is of the form $$ Q_{1} x_{1} Q_{2} x_{2} \cdots Q_{k} x_{k} P\left(x_{1}, x_{2}, \ldots, x_{k}\right) $$ where each \(Q_{i}, i=1,2, \ldots, k,\) is either the existential quantifier or the universal quantifier, and \(P\left(x_{1}, \ldots, x_{k}\right)\) is a predicate involving no quantifiers. For example, \(\exists x \forall y(P(x, y) \wedge Q(y))\) is in prenex normal form, whereas \(\exists x P(x) \vee \forall x Q(x)\) is not (because the quantifiers do not all occur first). Every statement formed from propositional variables, predicates, \(\mathbf{T},\) and \(\mathbf{F}\) using logical connectives and quantifiers is equivalent to a statement in prenex normal form. Exercise 51 asks for a proof of this fact. Show how to transform an arbitrary statement to a statement in prenex normal form that is equivalent to the given statement. (Note: A formal solution of this exercise requires use of structural induction, covered in Section \(5.3 . )\)

Prove that there are infinitely many solutions in positive integers \(x, y,\) and \(z\) to the equation \(x^{2}+y^{2}=\) \(z^{2} .\left[\text { Hint: Let } x=m^{2}-n^{2}, y=2 m n, \text { and } z=m^{2}+n^{2}\right.\) where \(m\) and \(n\) are integers. \(]\)

Translate each of these statements into logical expressions using predicates, quantifiers, and logical connectives. a) No one is perfect. b) Not everyone is perfect. c) All your friends are perfect. d) At least one of your friends is perfect. e) Everyone is your friend and is perfect. f) Not everybody is your friend or someone is not perfect.
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