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Chapter 1: Problem 39

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

Short Answer

Expert verified
Construct a table listing all combinations of truth values for p, q, and r, and evaluate the truth values of each compound proposition by breaking them down into simpler components step by step.

Step by step solution

01

Identify Variables

Determine the variables involved in each proposition. Each proposition here involves three variables: p, q, and r.
02

Create a Table Structure

Construct a table that includes columns for p, q, r, and the compound propositions.
03

List All Possible Truth Values

Write down all possible combinations of truth values for p, q, and r (8 combinations in total).
04

Evaluate \( eg q \)

Calculate the negation of q, \( eg q \), for each combination.
05

Evaluate \( eg q \vee r \)

Using the values of \( eg q \) and r, determine the truth value of \( eg q \vee r \) for each combination.
06

Evaluate \( p \rightarrow( eg q \vee r ) \)

Using the values of p and \( eg q \vee r \), calculate the truth value of \( p \rightarrow( eg q \vee r ) \).
07

Repeat for Other Propositions

Repeat Steps 4-6 for each proposition: \( eg p \rightarrow(q \rightarrow r), ((p \rightarrow q) \vee( eg p \rightarrow r)), ((p \rightarrow q) \wedge( eg p \rightarrow r)), ((p \rightarrow q) \vee( eg q \rightarrow r)), (( eg p \leftrightarrow eg q) \leftrightarrow(q \leftrightarrow r)) \).

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Key Concepts

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

Compound Propositions
In propositional logic, a compound proposition is a statement formed by combining one or more propositions using logical operators. For instance, if you have two simple propositions, 'p' and 'q', you can form compound propositions such as 'p AND q' or 'p OR q'. Compound propositions can be quite complex and involve multiple logical operators to form larger statements. Understanding how these propositions interact is crucial. Depending on the operators used, the truth value of a compound proposition can vary even if the truth values of the individual propositions remain the same. This is why constructing truth tables is a valuable tool in logic, as they help visualize how the different components of a compound proposition interact to produce a final truth value.
Logical Operators
Logical operators are symbols or words used to connect simple propositions and form compound propositions. Each operator has a specific logical function and affects the overall truth value of the compound proposition. Here are some common logical operators:
  • AND (鈭�): True if both propositions are true.
  • OR (鈭�): True if at least one of the propositions is true.
  • NOT (卢): Inverts the truth value of a proposition.
  • IMPLIES (鈫�): True unless the first proposition is true and the second is false.
  • EQUIVALENT (鈫�): True if both propositions have the same truth value.
For example, in the formula, 'p 鈭� q', the AND operator requires both 'p' and 'q' to be true for the entire expression to be true. Likewise, in 'p 鈫� q', the IMPLIES operator states that the expression is true except when 'p' is true and 'q' is false. By understanding how these operators work, you can better predict the behavior of compound propositions.
Truth Values
Truth values are the possible values a proposition can take: true (T) or false (F). When analyzing propositional logic, it's essential to consider all possible combinations of truth values for the involved variables. This is done through a truth table, which outlines every possible scenario for the propositions. For example, if you have three variables, 'p', 'q', and 'r', each can either be true or false. Hence, you have the following combinations:
  • T, T, T
  • T, T, F
  • T, F, T
  • T, F, F
  • F, T, T
  • F, T, F
  • F, F, T
  • F, F, F
By systematically going through these combinations, you can evaluate the truth value of a compound proposition under all possible scenarios. This process helps to ensure that the logic holds true across various conditions and isn't just accidentally true for a specific case.
Propositional Variables
Propositional variables are symbols representing simple propositions or statements that can either be true or false. Common symbols for propositional variables include 'p', 'q', 'r', and so forth. These variables serve as the building blocks for more complex logical expressions. When constructing a truth table or evaluating a logical expression, you assign truth values to these variables and analyze how they interact.
For instance, in the exercise above, 'p', 'q', and 'r' are propositional variables. Each can independently be either true or false, which then influences the truth values of the compound propositions they form. Understanding how these variables function and how to manipulate them using logical operators is fundamental to mastering propositional logic. By practicing with examples and constructing truth tables, you can get a better grasp of how different combinations of truth values for these variables affect the overall logic of the compound propositions.

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Most popular questions from this chapter

Exercises \(40-44\) deal with the translation between system specification and logical expressions involving quantifiers. Express each of these system specifications using predicates, quantifiers, and logical connectives. a) Every user has access to an electronic mailbox. b) The system mailbox can be accessed by everyone in the group if the file system is locked. c) The firewall is in a diagnostic state only if the proxy server is in a diagnostic state. d) At least one router is functioning normally if the throughput is between 100 kbps and 500 kbps and the proxy server is not in diagnostic mode.

Show that \(\exists x P(x) \wedge \exists x Q(x)\) and \(\exists x(P(x) \wedge Q(x))\) are not logically equivalent.

Use a proof by exhaustion to show that a tiling using dominoes of a \(4 \times 4\) checkerboard with opposite corners removed does not exist. [Hint: First show that you can assume that the squares in the upper left and lower right corners are removed. Number the squares of the original checkerboard from 1 to \(16,\) starting in the first row, moving right in this row, then starting in the leftmost square in the second row and moving right, and so on. Remove squares 1 and \(16 .\) To begin the proof, note that square 2 is covered either by a domino laid horizontally, which covers squares 2 and \(3,\) or vertically, which covers squares 2 and \(6 .\) Consider each of these cases separately, and work through all the subcases that arise. \(]\)

Exercises \(61-64\) are based on questions found in the book Symbolic Logic by Lewis Carroll. Let P(x), Q(x), R(x), and S(x) be the statements 鈥渪 is a duck,鈥� 鈥渪 is one of my poultry,鈥� 鈥渪 is an officer,鈥� and 鈥渪 is willing to waltz,鈥� respectively. Express each of these statements using quantifiers; logical connectives; and P(x), Q(x), R(x), and S(x). a) No ducks are willing to waltz. b) No officers ever decline to waltz. c) All my poultry are ducks. d) My poultry are not officers. e) Does (d) follow from (a), (b), and (c)? If not, is there a correct conclusion?

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