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Chapter 1: Problem 18

Write truth tables for the statement forms in \(14-18\). $$ (p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r) $$

Short Answer

Expert verified
The short answer for the truth table of the given statement form \((p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r)\) is: $$ \begin{array}{c|c|c|c} p & q & r & ((p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r)) \\ \hline T & T & T & T \\ T & T & F & F \\ T & F & T & T \\ T & F & F & T \\ F & T & T & T \\ F & T & F & F \\ F & F & T & T \\ F & F & F & T \end{array} $$

Step by step solution

01

Write down all possible truth value combinations

We have three variables \(p\), \(q\), and \(r\). The first step is to create a table with all possible combinations of the truth values for these variables. $$ \begin{array}{c|c|c} p & q & r \\ \hline T & T & T \\ T & T & F \\ T & F & T \\ T & F & F \\ F & T & T \\ F & T & F \\ F & F & T \\ F & F & F \end{array} $$
02

Calculate the truth values for sub-expressions

Now, we need to calculate the truth values for each sub-expression in the statement. - \(\sim p\): This is the negation of \(p\). - \(\sim q\): This is the negation of \(q\). - \(\sim r\): This is the negation of \(r\). - \((\sim p \vee q)\): This is the disjunction of \(\sim p\) and \(q\). - \((p \vee (\sim p \vee q))\): This is the disjunction of \(p\) and \((\sim p \vee q)\). - \((q \wedge \sim r)\): This is the conjunction of \(q\) and \(\sim r\). - \(\sim (q \wedge \sim r)\): This is the negation of \((q \wedge \sim r)\). - \(((p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r))\): This is the conjunction of \((p \vee (\sim p \vee q))\) and \(\sim (q \wedge \sim r)\).
03

Complete the truth table

Finally, we can write the complete truth table using the information we have gathered in steps 1 and 2. $$ \begin{array}{c|c|c|c|c|c|c|c|c} p & q & r & \sim p & \sim q & \sim r & (\sim p \vee q) & (p \vee (\sim p \vee q)) & (q \wedge \sim r) & \sim(q \wedge \sim r) & ((p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r)) \\ \hline T & T & T & F & F & F & T & T & F & T & T \\ T & T & F & F & F & T & T & T & T & F & F \\ T & F & T & F & T & F & F & T & F & T & T \\ T & F & F & F & T & T & F & T & F & T & T \\ F & T & T & T & F & F & T & T & F & T & T \\ F & T & F & T & F & T & T & T & T & F & F \\ F & F & T & T & T & F & T & T & F & T & T \\ F & F & F & T & T & T & T & T & F & T & T \end{array} $$

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Key Concepts

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

Logical Expressions
Logical expressions are the building blocks of truth tables. They are made up of variables and logical operators such as negation, conjunction, and disjunction. In the exercise, we see an expression like \((p \vee(\sim p \vee q)) \wedge \sim(q \wedge \sim r)\). This expression combines variables \(p\), \(q\), and \(r\) with different logical operations. Understanding how these operators work together helps us evaluate the truth value for each possible scenario. Logical expressions are essential for constructing truth tables as they determine how combinations of variables interact with one another.
Negation
Negation is one of the primary logical operators, represented by the symbol \(\sim\). It simply reverses the truth value of a given variable. For example, if a variable \(p\) is true (T), then its negation \(\sim p\) is false (F), and vice versa. In the given exercise, negations are used in various components like \(\sim p\), \(\sim q\), and \(\sim r\). Negation is crucial because it allows us to consider the opposite scenario for each variable, which is fundamental in evaluating all possible outcomes in a truth table.
Conjunction
Conjunction is another key logical operator, symbolized by \(\wedge\). A conjunction statement is only true if both of its operands are true. In the expression \((q \wedge \sim r)\), the conjunction combines \(q\) and the negation of \(r\) (\(\sim r\)).
This part is true only when \(q\) is true and \(\sim r\) is true (meaning \(r\) is false). Otherwise, the result is false. This highlights how conjunction demands all conditions be met, making it stricter than other logical operations like disjunction.
Disjunction
Disjunction is represented by the symbol \(\vee\). It evaluates to true if at least one of its operands is true. For instance, in the exercise, disjunction appears in expressions like \((\sim p \vee q)\) and \((p \vee (\sim p \vee q))\).
The first part \((\sim p \vee q)\) is true if either \(\sim p\) (the negation of \(p\)) is true or \(q\) is true. Similarly, the larger expression combines these with \(p\), and it remains true if any of these conditions are met. Disjunction thus offers flexibility, needing at least one true element to confirm the truth of the complete expression.

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Most popular questions from this chapter

Write each of the following three statements in symbolic form and determine which pairs are logically equivalent. Include truth tables and a few words of explanation. If it walks like a duck and it talks like a duck, then it is a duck. Either it does not walk like a duck or it does not talk like a duck, or it is a duck. If it does not walk like a duck and it does not talk like a duck, then it is not a duck.

A conditional statement and its contrapositive are logically equivalent to each other.

Assume \(x\) is a particular real number and use De Morgan's laws to write negations for the statements in 35-38. $$ -2

"Do you mean that you think you can find out the answer to it?" said the March Hare. "Exactly so," said Alice. "Then you should say what you mean," the March Hare went on. "I do," Alice hastily replied; "at least-at least I mean what I say - that's the same thing, you know." "Not the same thing a bit"" said the Hatter. "Why, you might just as well say that 'I see what I eat' is the same thing as 'I eat what I see'!" \- from "A Mad Tea-Party" in Alice in Wonderland, by Lewis Carroll The Hatter is right. "I say what I mean" is not the same thing as "I mean what I say." Rewrite each of these two sentences in if-then form and explain the logical relation between them. (This exercise is referred to in the introduction to Chapter 3.)

Some of the arguments in 24-32 are valid, whereas others exhibit the converse or the inverse error. Use symbols to write the logical form of each argument. If the argument is valid, identify the rule of inference that guarantees its validity. Otherwise, state whether the converse or the inverse error is made. If this number is larger than 2, then its square is larger than 4 . This number is not larger than 2 . \(\therefore\) The square of this number is not larger than 4 .
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