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Chapter 1: Problem 41

Construct a truth table for \((p \leftrightarrow q) \leftrightarrow(r \leftrightarrow s)\)

Short Answer

Expert verified
Evaluate \(p \leftrightarrow q \leftrightarrow r \leftrightarrow s\) for all 16 combinations of p, q, r, and s

Step by step solution

01

- Identify Variables

Identify all the variables in the expression. Here, the variables are p, q, r, and s.
02

- List All Possible Truth Values

List all possible combinations of truth values for the variables. There are four variables, so there will be a total of 2^4 = 16 combinations.
03

- Compute Intermediate Expression 1

For each combination of truth values, compute the truth value of the expression \(p \leftrightarrow q\). This will produce an intermediate column in the truth table.
04

- Compute Intermediate Expression 2

Next, for each combination of truth values, compute the truth value of the expression \(r \leftrightarrow s\). This will produce another intermediate column in the truth table.
05

- Compute Final Expression

Finally, compute the truth value of the entire expression \((p \leftrightarrow q) \leftrightarrow (r \leftrightarrow s)\) for each combination of truth values by comparing the earlier intermediate results.
06

- Complete the Truth Table

Fill out the truth table by placing the results of the intermediate expressions and the final expression in the corresponding columns. The completed truth table should show the truth values of the entire expression for each possible combination of p, q, r, and s.

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Key Concepts

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

Logical Expressions
Logical expressions use variables and logical operators to create statements that can be either true or false. These operators include AND (\( \land \)), OR (\( \lor \)), NOT (\( eg \)), and the biconditional (\( \leftrightarrow \)).
In our exercise, we use four variables: p, q, r, and s. Each of these variables can be either true or false.
Logical expressions help in forming statements that can be evaluated systematically.
This is crucial for constructing truth tables.
Truth Tables
Truth tables are a great way to visualize all possible truth values of a logical expression. They show how the truth value of a complex expression is derived from its components.
To construct a truth table:
  • List all variables.
  • Determine all possible truth value combinations.
  • Evaluate the expression step by step.

In our task, we have 4 variables, creating 16 combinations. Each combination helps in evaluating the final truth of the expression.
Biconditional Operator
The biconditional operator (\( \leftrightarrow \)) evaluates whether two statements are logically equivalent. It is true when both statements are either true or false and false otherwise.
For example:
  • \( p \leftrightarrow q \) is true if both p and q are true or both are false.
  • \( p \leftrightarrow q \) is false if one is true and the other is false.

In our exercise, we first evaluate \( p \leftrightarrow q \) and \( r \leftrightarrow s \) separately. Then, we compare these intermediate results using the biconditional operator to get the final expression: \( (p \leftrightarrow q) \leftrightarrow (r \leftrightarrow s) \).
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Logic and truth tables are core components.
In discrete math, we deal with finite sets of elements and non-continuous functions.
Key topics include:
  • Set Theory
  • Combinatorics
  • Graph Theory
  • Logic

Understanding logical expressions and truth tables equips students with the skills to navigate more complex problems within discrete mathematics.

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Most popular questions from this chapter

Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. a) Someone in your class can speak Hindi. b) Everyone in your class is friendly. c) There is a person in your class who was not born in California. d) A student in your class has been in a movie. e) No student in your class has taken a course in logic programming.

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

Let \(C(x)\) be the statement " \(x\) has a cat," let \(D(x)\) be the statement " \(x\) has a dog," and let \(F(x)\) be the statement "x has a ferret." Express each of these statements in terms of \(C(x), D(x), F(x),\) quantifiers, and logical connectives. Let the domain consist of all students in your class. a) A student in your class has a cat, a dog, and a ferret. b) All students in your class have a cat, a dog, or a ferret. c) Some student in your class has a cat and a ferret, but not a dog. d) No student in your class has a cat, a dog, and a ferret. e) For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet.

Use rules of inference to show that if \(\forall x(P(x) \vee Q(x))\) and \(\forall x((\neg P(x) \wedge Q(x)) \rightarrow R(x))\) are true, then \(\forall x(\neg R(x) \rightarrow\) \(P(x)\) is also true, where the domains of all quantifiers are the same.

Find a counterexample, if possible, to these universally quantified statements, where the domain for all variables consists of all integers. a) \(\forall x \forall y\left(x^{2}=y^{2} \rightarrow x=y\right)\) b) \(\forall x \exists y\left(y^{2}=x\right)\) c) \(\forall x \forall y(x y \geq x)\)
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