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Chapter 1: Problem 23

Each of Exercises \(20-32\) asks you to show that two compound propositions are logically equivalent. To do this, either show that both sides are true, or that both sides are false, for exactly the same combinations of truth values of the propositional variables in these expressions (whichever is easier). Show that \(\neg p \leftrightarrow q\) and \(p \leftrightarrow \neg q\) are logically equivalent.

Short Answer

Expert verified
The propositions \(eg p \leftrightarrow q\) and \(p \leftrightarrow eg q\) have the same outputs for all truth values, hence they are logically equivalent.

Step by step solution

01

- Define the compound propositions

Define the given compound propositions for reference:1. Proposition 1: \(eg p \leftrightarrow q\)2. Proposition 2: \(p \leftrightarrow eg q\).
02

- Write a truth table for both propositions

Create a truth table considering all possible truth values of \(p\) and \(q\).table:\begin{array}{|c|c|c|c|c|c|}\hline p & q & eg p & eg q & (eg p \leftrightarrow q) & (p \leftrightarrow eg q) \hlineT & T & F & F & F & F \T & F & F & T & T & T \F & T & T & F & T & T \F & F & T & T & F & F \hline\end{array}
03

- Determine the output for each proposition

Use the truth table to compare outputs for each case:1. For the first row (T, T), both \(eg p \leftrightarrow q\) and \(p \leftrightarrow eg q \) are F.2. For the second row (T, F), both expressions are T.3. For the third row (F, T), both expressions are T.4. For the fourth row (F, F), both expressions are F.
04

- Compare the outputs

Observe that the outputs for \(eg p \leftrightarrow q\) and \(p \leftrightarrow eg q\) are the same for all possible combinations of truth values for \(p\) and \(q\).

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Key Concepts

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

Truth Table
A truth table is a mathematical table used in logic to determine the truth values of logical expressions based on their components. In our exercise, we create a truth table to compare two compound propositions. A truth table has columns representing all possible truth values of the propositional variables and the resultant truth values of the logical expressions for each combination.

For two propositions, say \( p \) and \( q \), we have four combinations of truth values:
  • \( p = T \), \( q = T \)
  • \( p = T \), \( q = F \)
  • \( p = F \), \( q = T \)
  • \( p = F \), \( q = F \)
Using these combinations, we evaluate our compound propositions \( eg p \leftrightarrow q \) and \( p \leftrightarrow eg q \). For each row (combination of truth values), we assess the results of these propositions to check whether they produce the same truth values, which can show their logical equivalence.
Propositional Logic
Propositional logic, also known as propositional calculus, is a branch of logic that deals with propositions and their logical relationships and connections via operators. Propositions are statements that can either be true or false but not both simultaneously.

In propositional logic, we use symbols to represent propositions (like \( p \) and \( q \)). Logical operators such as AND (\( \land \)), OR (\( \/ \)), NOT (\( eg \)), and IMPLIES (\( -> \)) connect these propositions to form compound statements.

For our exercise, we use the biconditional operator (\( \leftrightarrow \)), which states that two propositions are true if they are either both true or both false. By exploring the propositions \( eg p \leftrightarrow q \) and \( p \leftrightarrow eg q \), we examine their truth values under various conditions and use their outputs to determine logical equivalence.
Compound Propositions
Compound propositions are formed by combining two or more simpler propositions using logical operators. They allow us to make more complex statements that can convey richer information.

Given two simple propositions \( p \) and \( q \), we can form the compound propositions \( eg p \leftrightarrow q \) and \( p \leftrightarrow eg q \). Here, \( eg p \) represents the negation of \( p \), and \( eg q \) is the negation of \( q \).

The biconditional statement \( A \leftrightarrow B \) means that \( A \) and \( B \) must have the same truth value. Therefore, for \( eg p \leftrightarrow q \) and \( p \leftrightarrow eg q \) to be equivalent, they must yield the same truth values across all possible combinations of \( p \) and \( q \). By constructing a truth table and comparing the outputs of these propositions for every combination, we determine if they are logically equivalent.

Understanding compound propositions is crucial in evaluating complex logical statements and proving logical equivalence as illustrated in the exercise.

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Most popular questions from this chapter

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) Some old dogs can learn new tricks. b) No rabbit knows calculus. c) Every bird can fly. d) There is no dog that can talk. e) There is no one in this class who knows French and Russian.

The Logic Problem, taken from \(W F F^{\prime} N\) PROOF, The Game of Logic, has these two assumptions: 1\. "Logic is difficult or not many students like logic." 2\. "If mathematics is easy, then logic is not difficult." By translating these assumptions into statements involving propositional variables and logical connectives, de- termine whether each of the following are valid conclusions of these assumptions: a) That mathematics is not easy, if many students like logic. b) That not many students like logic, if mathematics is not easy. c) That mathematics is not easy or logic is difficult. d) That logic is not difficult or mathematics is not easy. e) That if not many students like logic, then either mathematics is not easy or logic is not difficult.

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Translate these specifications into English, where \(F(p)\) is "printer \(p\) is out of service," \(B(p)\) is "Printer \(p\) is busy," \(L(j)\) is "Print job \(j\) is lost, " and \(Q(j)\) is "Print job \(j\) is queued." $$ \begin{array}{l}{\text { a) } \exists p(F(p) \wedge B(p)) \rightarrow \exists j L(j)} \\ {\text { b) } \forall p B(p) \rightarrow \exists j Q(j)} \\\ {\text { c) } \exists j(Q(j) \wedge L(j)) \rightarrow \exists p F(p)} \\\ {\text { d) }(\forall p B(p) \wedge \forall j Q(j)) \rightarrow \exists j L(j)}\end{array} $$

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 鈥淚t is not the case that.鈥�) a) Every student in this class has taken exactly two mathematics classes at this school. b) Someone has visited every country in the world except Libya. c) No one has climbed every mountain in the Himalayas. d) Every movie actor has either been in a movie with Kevin Bacon or has been in a movie with someone who has been in a movie with Kevin Bacon.

Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives). a) \(\neg \forall x \forall y P(x, y) \quad\) b) \(\neg \forall y \exists x P(x, y)\) c) \(\neg \forall y \forall x(P(x, y) \vee Q(x, y))\) d) \(\neg(\exists x \exists y \neg P(x, y) \wedge \forall x \forall y Q(x, y))\) e) \(\quad \neg \forall x(\exists y \forall z P(x, y, z) \wedge \exists z \forall y P(x, y, z))\)
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