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Chapter 1: Problem 38

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

Short Answer

Expert verified
The truth tables for each proposition are:(a) (p 鈭 q) 鈭 r(b) (p 鈭 q) 鈭 r(c) (p 鈭 q) 鈭 r(d) (p 鈭 q) 鈭 r(e) (p 鈭 q) 鈭 卢r(f) (p 鈭 q) 鈭 卢r

Step by step solution

01

- List All Possible Truth Values

There are three variables: p, q, and r. List out all the possible combinations of truth values (True or False) for these variables. There will be 2^3 = 8 combinations:1. p = T, q = T, r = T2. p = T, q = T, r = F3. p = T, q = F, r = T4. p = T, q = F, r = F5. p = F, q = T, r = T6. p = F, q = T, r = F7. p = F, q = F, r = T8. p = F, q = F, r = F
02

- Construct Truth Table for (p 鈭 q) 鈭 r

To construct the truth table for \( (p \vee q) \vee r \) we need intermediate columns for \( p \vee q \):| p | q | r | p 鈭 q | (p 鈭 q) 鈭 r ||---|---|---|-------|-------------|| T | T | T | T | T || T | T | F | T | T || T | F | T | T | T || T | F | F | T | T || F | T | T | T | T || F | T | F | T | T || F | F | T | F | T || F | F | F | F | F |
03

- Construct Truth Table for (p 鈭 q) 鈭 r

To construct the truth table for \( (p \vee q) \wedge r \) we need intermediate columns for \( p \vee q \):| p | q | r | p 鈭 q | (p 鈭 q) 鈭 r ||---|---|---|-------|-------------|| T | T | T | T | T || T | T | F | T | F || T | F | T | T | T || T | F | F | T | F || F | T | T | T | T || F | T | F | T | F || F | F | T | F | F || F | F | F | F | F |
04

- Construct Truth Table for (p 鈭 q) 鈭 r

To construct the truth table for \( (p \wedge q) \vee r \) we need intermediate columns for \( p \wedge q \):| p | q | r | p 鈭 q | (p 鈭 q) 鈭 r ||---|---|---|-------|-------------|| T | T | T | T | T || T | T | F | T | T || T | F | T | F | T || T | F | F | F | F || F | T | T | F | T || F | T | F | F | F || F | F | T | F | T || F | F | F | F | F |
05

- Construct Truth Table for (p 鈭 q) 鈭 r

To construct the truth table for \( (p \wedge q) \wedge r \) we need intermediate columns for \( p \wedge q \):| p | q | r | p 鈭 q | (p 鈭 q) 鈭 r ||---|---|---|-------|-------------|| T | T | T | T | T || T | T | F | T | F || T | F | T | F | F || T | F | F | F | F || F | T | T | F | F || F | T | F | F | F || F | F | T | F | F || F | F | F | F | F |
06

- Construct Truth Table for (p 鈭 q) 鈭 卢r

To construct the truth table for \( (p \vee q) \wedge \eg r \), we need intermediate columns for \( p \vee q \) and \( \eg r \):| p | q | r | p 鈭 q | \eg r | (p 鈭 q) 鈭 卢r ||---|---|---|-------|-------|---------------|| T | T | T | T | F | F || T | T | F | T | T | T || T | F | T | T | F | F || T | F | F | T | T | T || F | T | T | T | F | F || F | T | F | T | T | T || F | F | T | F | F | F || F | F | F | F | T | F |
07

- Construct Truth Table for (p 鈭 q) 鈭 卢r

To construct the truth table for \( (p \wedge q) \vee \eg r \), we need intermediate columns for \( p \wedge q \) and \( \eg r \):| p | q | r | p 鈭 q | \eg r | (p 鈭 q) 鈭 卢r ||---|---|---|-------|-------|---------------|| T | T | T | T | F | T || T | T | F | T | T | T || T | F | T | F | F | F || T | F | F | F | T | T || F | T | T | F | F | F || F | T | F | F | T | T || F | F | T | F | F | F || F | F | F | F | T | T |

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Key Concepts

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

Propositional Logic
Propositional logic is a branch of logic that deals with propositions, which are statements that can be either true or false. In this type of logic, propositions are represented by variables such as \(p\), \(q\), and \(r\). Each of these variables can take the value of either true (T) or false (F). Understanding propositional logic is essential because it forms the foundation for reasoning and computational logic.
For example, the statement 'It is raining' can be a proposition. If it is indeed raining, the proposition is true. If not, the proposition is false. Combining multiple propositions using logical connectives allows us to create more complex logical statements called compound propositions.
Truth Tables
Truth tables are a useful tool in propositional logic to show all possible truth values of a compound proposition. Each row of the truth table represents a specific combination of truth values for the variables involved. This helps in understanding how the truth value of the compound proposition is determined based on the truth values of its constituent propositions.
To construct a truth table:
  • List all possible combinations of truth values for the variables.
  • Determine the truth values of the intermediate expressions.
  • Combine these to find the final truth value of the compound proposition.
In our given problem, there are three variables: \(p\), \(q\), and \(r\), resulting in 8 combinations (because there are \(2^3\) possible combinations of true and false). Each truth table will show how these combinations affect the final outcome.
Logical Connectives
Logical connectives are symbols used to connect propositions and form compound propositions. The primary logical connectives are:
  • \(\vee\) (OR): True if at least one of the operands is true.
  • \(\wedge\) (AND): True only if both operands are true.
  • \(eg\) (NOT): Inverts the truth value of the operand.
These connectives help in forming more complex logical expressions. For instance:
- The expression \(p \vee q\) (p OR q) is true if either \(p\) or \(q\) is true.
- The expression \(p \wedge q\) (p AND q) is true only if both \(p\) and \(q\) are true.
- The expression \(eg p\) (NOT p) is true if \(p\) is false, and vice versa.
Understanding these connectives and their impact on truth values is crucial for constructing accurate truth tables and ensuring correct logical reasoning.
Compound Propositions
Compound propositions are statements formed by combining two or more propositions using logical connectives. These compound propositions can be evaluated based on the truth values of their constituent propositions.
Let's consider some of the examples from the exercise:
  • \((p \vee q) \vee r\) is a compound proposition using the OR connective.
  • \((p \vee q) \wedge r\) uses both OR and AND connectives.
  • \((p \wedge q) \vee r\) combines propositions with AND first, then OR.
  • \((p \wedge q) \wedge r\) uses the AND connective twice.
  • \((p \vee q) \wedge eg r\) combines OR, AND, and NOT connectives.
  • \((p \wedge q) \vee eg r\) uses AND, OR, and NOT connectives.
By creating a truth table for each compound proposition, you can systematically determine the truth value of the compound statement for every possible combination of truth values of its parts.
This not only aids in verifying logical expressions but also enhances understanding of how complex logical statements are built and evaluated.

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Most popular questions from this chapter

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

Suppose the domain of the propositional function \(P(x, y)\) consists of pairs \(x\) and \(y,\) where \(x\) is \(1,2,\) or 3 and \(y\) is \(1,2,\) or \(3 .\) Write out these propositions using disjunctions and conjunctions. $$ \begin{array}{ll}{\text { a) } \forall x \forall y P(x, y)} & {\text { b) } \exists x \exists y P(x, y)} \\ {\text { c) } \exists x \forall y P(x, y)} & {\text { d) } \forall y \exists x P(x, y)}\end{array} $$

Let M(x, y) be 鈥渪 has sent y an e-mail message鈥 and T(x, y) be 鈥渪 has telephoned y,鈥 where the domain consists of all students in your class. Use quantifiers to express each of these statements. (Assume that all e-mail messages that were sent are received, which is not the way things often work.) a) Chou has never sent an e-mail message to Koko. b) Arlene has never sent an e-mail message to or tele- phoned Sarah. c) Jose has never received an e-mail message from Deborah. d) Every student in your class has sent an e-mail mes- sage to Ken. e) No one in your class has telephoned Nina. f ) Everyone in your class has either telephoned Avi or sent him an e-mail message. g) There is a student in your class who has sent every- one else in your class an e-mail message. h) There is someone in your class who has either sent an e-mail message or telephoned everyone else in your class. i) There are two different students in your class who have sent each other e-mail messages. j) There is a student who has sent himself or herself an e-mail message. k) There is a student in your class who has not received an e-mail message from anyone else in the class and who has not been called by any other student in the class. l) Every student in the class has either received an email message or received a telephone call from another student in the class. m) There are at least two students in your class such that one student has sent the other e-mail and the second student has telephoned the first student. n) There are two different students in your class who between them have sent an e-mail message to or telephoned everyone else in the class.

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
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