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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

Show that if \(a\) and \(b\) are integers in the range 1 through 128 , and the sum of \(a\) and \(b\) is also in this range, then \(2^{8} \leq\left(2^{8}-a\right)+\left(2^{8}-b\right)<2^{9}\). Explain why it follows that the binary representation of \(\left(2^{8}-a\right)+\left(2^{8}-b\right)\) has a leading 1 in the \(2^{8}\) th position.

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

Use truth tables to determine whether the argument forms in 6-10 are valid. Indicate which columns represent the premises and which represent the conclusion, and include a few words of explanation to support your answers. $$ \begin{aligned} & p \\ & p \rightarrow q \\ & \sim q \vee r \\ \therefore & r \end{aligned} $$

"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.)

Explain in your own words what distinguishes a valid form of argument from an invalid one.
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