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Chapter 1: Problem 10

Let \(p\) and \(q\) be the propositions \(p :\) I bought a lottery ticket this week. \(q :\) I won the million dollar jackpot. Express each of these propositions as an English sentence. a) \(\neg p\) b) \(p \vee q\) c) \(p \rightarrow q\) d) \(p \wedge q\) e) \(p \leftrightarrow q\) f) \(\neg p \rightarrow \neg q\) g) \(\neg p \wedge \neg q\) h) \(\neg p \vee(p \wedge q)\)

Short Answer

Expert verified
(a) I did not buy a lottery ticket this week. (b) I bought a lottery ticket this week or I won the million dollar jackpot. (c) If I bought a lottery ticket this week, then I won the million dollar jackpot. (d) I bought a lottery ticket this week and I won the million dollar jackpot. (e) I bought a lottery ticket this week if and only if I won the million dollar jackpot. (f) If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot. (g) I did not buy a lottery ticket this week and I did not win the million dollar jackpot. (h) I did not buy a lottery ticket this week or (I bought a lottery ticket this week and I won the million dollar jackpot).

Step by step solution

01

Interpret the given propositions

Given propositions are: - \( p : \text{I bought a lottery ticket this week.} \) - \( q : \text{I won the million dollar jackpot.} \)
02

Translate each logical expression to English

Express each logical expression as an English sentence.
03

Translate \( eg p \)

\( eg p \) means 'not p', so it translates to 'I did not buy a lottery ticket this week.'
04

Translate \( p \vee q \)

\( p \vee q \) means 'p or q' (inclusive or), so it translates to 'I bought a lottery ticket this week or I won the million dollar jackpot.'
05

Translate \( p \rightarrow q \)

\( p \rightarrow q \) means 'if p then q', so it translates to 'If I bought a lottery ticket this week, then I won the million dollar jackpot.'
06

Translate \( p \wedge q \)

\( p \wedge q \) means 'p and q', so it translates to 'I bought a lottery ticket this week and I won the million dollar jackpot.'
07

Translate \( p \leftrightarrow q \)

\( p \leftrightarrow q \) means 'p if and only if q', so it translates to 'I bought a lottery ticket this week if and only if I won the million dollar jackpot.'
08

Translate \( eg p \rightarrow eg q \)

\( eg p \rightarrow eg q \) means 'if not p then not q', so it translates to 'If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot.'
09

Translate \( eg p \wedge eg q \)

\( eg p \wedge eg q \) means 'not p and not q', so it translates to 'I did not buy a lottery ticket this week and I did not win the million dollar jackpot.'
10

Translate \( eg p \vee (p \wedge q) \)

\( eg p \vee (p \wedge q) \) means 'not p or (p and q)', so it translates to 'I did not buy a lottery ticket this week or (I bought a lottery ticket this week and I won the million dollar jackpot).'

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Key Concepts

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

Propositional Logic
Propositional logic, also known as statement logic or sentential logic, is the branch of logic that deals with propositions, which can either be true or false. It focuses on the ways in which propositions can be combined and manipulated using logical connectives. These logical connectives include:
  • Conjunction (\(\wedge\)) - which means 'and'
  • Disjunction (\(\vee\)) - which means 'or'
  • Negation (\(eg\)) - which means 'not'
  • Implication (\(\rightarrow\)) - which means 'if... then...'
  • Biconditional (\(\leftrightarrow\)) - which means 'if and only if'
Each logical connective helps to build complex logical statements from simpler ones. The truth value of these complex statements is determined by the truth values of their component propositions.
Translation of Logical Expressions
Translating logical expressions into English sentences involves understanding the meaning of each logical operator used. Let's use the provided exercise as an example: Given two propositions,
p : I bought a lottery ticket this week.
q : I won the million dollar jackpot.
Various logical expressions and their translations are as follows:
  • \( eg p \) translates to 'I did not buy a lottery ticket this week.'
  • \( p \vee q \) translates to 'I bought a lottery ticket this week or I won the million dollar jackpot.'
  • \( p \rightarrow q \) translates to 'If I bought a lottery ticket this week, then I won the million dollar jackpot.'
  • \( p \wedge q \) translates to 'I bought a lottery ticket this week and I won the million dollar jackpot.'
  • \( p \leftrightarrow q \) translates to 'I bought a lottery ticket this week if and only if I won the million dollar jackpot.'
  • \( eg p \rightarrow eg q \) translates to 'If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot.'
  • \( eg p \wedge eg q \) translates to 'I did not buy a lottery ticket this week and I did not win the million dollar jackpot.'
  • \( eg p \vee (p \wedge q) \) translates to 'I did not buy a lottery ticket this week or (I bought a lottery ticket this week and I won the million dollar jackpot).'
By consistently applying these translation rules, it's possible to convert any logical expression into a clear English sentence.
Logical Operators
Logical operators are symbols or words used to connect propositions in ways that form new propositions. Understanding each of these operators is key to mastering propositional logic:
  • Negation (\(eg\)) - Negates or reverses the truth value of a proposition. If p is true, \(eg p\) is false, and vice versa.
  • Conjunction (\(\wedge\)) - Represents 'and'. The conjunction of propositions p and q, written as \(p \wedge q\), is true only if both p and q are true.
  • Disjunction (\(\vee\)) - Represents 'or'. The disjunction of propositions p and q, written as \(p \vee q\), is true if either p is true, q is true, or both are true.
  • Implication (\(\rightarrow\)) - Represents 'if... then...'. The implication \(p \rightarrow q\) is false only if p is true and q is false; otherwise, it is true.
  • Biconditional (\(\leftrightarrow\)) - Represents 'if and only if'. The biconditional \(p \leftrightarrow q\) is true if both p and q are either true or false.
By combining these operators with propositions, complex logical statements can be formed. Understanding their use and impact on the truth values of propositions is essential for translating and working with logical expressions effectively.

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Most popular questions from this chapter

Prove that given a real number \(x\) there exist unique numbers \(n\) and \(\epsilon\) such that \(x=n+\epsilon, n\) is an integer, and \(0 \leq \epsilon<1 .\)

For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) 鈥淚f I play hockey, then I am sore the next day.鈥� 鈥淚 use the whirlpool if I am sore.鈥� 鈥淚 did not use the whirlpool.鈥� b) 鈥淚f I work, it is either sunny or partly sunny.鈥� 鈥淚 worked last Monday or I worked last Friday.鈥� 鈥淚t was not sunny on Tuesday.鈥� 鈥淚t was not partly sunny on Friday.鈥� c) 鈥淎ll insects have six legs.鈥� 鈥淒ragonflies are insects.鈥� 鈥淪piders do not have six legs.鈥� 鈥淪piders eat dragon-flies.鈥� d) 鈥淓very student has an Internet account.鈥� 鈥淗omer does not have an Internet account.鈥� 鈥淢aggie has an Internet account.鈥� e) 鈥淎ll foods that are healthy to eat do not taste good.鈥� 鈥淭ofu is healthy to eat.鈥� 鈥淵ou only eat what tastes good.鈥� 鈥淵ou do not eat tofu.鈥� 鈥淐heeseburgers are not healthy to eat.鈥� f ) 鈥淚 am either dreaming or hallucinating.鈥� 鈥淚 am not dreaming.鈥� 鈥淚f I am hallucinating, I see elephants running down the road.鈥�

Express the negations of each of these statements so that all negation symbols immediately precede predicates. a) \(\forall x \exists y \forall z T(x, y, z)\) b) \(\forall x \exists y P(x, y) \vee \forall x \exists y Q(x, y)\) c) \(\forall x \exists y P(x, y) \wedge \exists z R(x, y, z) )\) d) \(\forall x \exists y(P(x, y) \rightarrow Q(x, y))\)

Determine whether these are valid arguments. a) If \(x\) is a positive real number, then \(x^{2}\) is a positive real number. Therefore, if \(a^{2}\) is positive, where \(a\) is a real number, then \(a\) is a positive real number. b) If \(x^{2} \neq 0,\) where \(x\) is a real number, then \(x \neq 0 .\) Let \(a\) be a real number with \(a^{2} \neq 0 ;\) then \(a \neq 0\)

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