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Chapter 1: Problem 24

Write the truth table of each proposition. $$ (\neg p \vee \neg q) \vee p $$

Short Answer

Expert verified
The truth table is as follows: (with T for true, F for false) \n \[\begin{array}{|c||c|c|c|c|}\hlinep & q & \neg p & \neg q & (\neg p \vee \neg q) \vee p \\hlineT & T & F & F & T \T & F & F & T & T \F & T & T & F & T \F & F & T & T & T \\hline\end{array}\]

Step by step solution

01

Understand The Proposition

This given proposition is \((\neg p \vee \neg q) \vee p\). This is a logical formula involving the logical variables 'p' and 'q'. The logical connectives are OR \((\vee)\) and NOT \((\neg)\). The NOT operation is performed before OR operation due to precedence rules.
02

Set Up Basic Truth Table

First, set up a basic truth table for 'p' and 'q'. Since each of 'p' and 'q' can either be true (T) or false (F), we have a total of \(2^2 = 4\) rows in the truth table, representing all possibilities.
03

Find \(\neg p\) and \(\neg q\)

Perform NOT operation on 'p' and 'q'. NOT operation converts true (T) to false (F) and vice versa.
04

Calculate \((\neg p \vee \neg q)\)

Perform OR operation on \(\neg p\) and \(\neg q\). OR operation is true whenever either (or both) of its component propositions are true.
05

Evaluate The Whole Proposition

Lastly, find the truth value of the whole proposition \((\neg p \vee \neg q) \vee p\). This is obtained by performing OR operation on steps 3 and 4.

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Key Concepts

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

Logical Connectives
Logical connectives are symbols or words used to connect propositions in propositional logic. They are essential building blocks for constructing complex statements from simpler ones.
One of the primary logical connectives is OR, represented by the symbol \(\vee\). The OR connective is used to join two propositions, and the result is true if at least one of the propositions is true.
Another important connective is NOT, represented by \(eg\). It operates on a single proposition, reversing its truth value: if the proposition is true, NOT makes it false, and vice versa.
  • These connectives follow specific precedence rules, meaning NOT is processed before OR, just like multiplication is processed before addition in arithmetic.
  • Understanding logical connectives is crucial when working with truth tables, as they determine how you interpret and combine the values of propositions.
By understanding the role and function of logical connectives, students can approach complex logical formulas with greater clarity and confidence.
Propositional Logic
Propositional logic is a branch of logic that deals with propositions and their interconnections using logical connectives.
A proposition is a statement that can be either true or false but not both simultaneously. In propositional logic, we use variables like 'p' and 'q' to represent propositions.
  • Complex propositions can be formed by combining simple propositions using logical connectives such as AND, OR, and NOT.
  • These connectives form the syntax of propositional logic and allow us to express logical relationships.
  • A truth table is an invaluable tool in propositional logic, allowing you to systematically list all possible truth values for a given expression.
The goal of propositional logic is to determine the truth value of complex propositions based on the truth values of their constituent simple propositions.
By mastering propositional logic, you will be equipped to break down and evaluate logical statements efficiently.
Logical Operations
Logical operations are processes that apply logical connectives to propositions, transforming or combining them to form new propositions.
These operations are at the heart of propositional logic, allowing us to manipulate and derive conclusions from given propositions.
  • The OR operation, symbolized by \(\vee\), states that if any of the individual propositions is true, then the combined proposition is true.
  • The NOT operation \(eg\), inverses the truth value of a single proposition, turning true arrangements into false and vice versa.
  • In the given proposition, the operations include using NOT to negate 'p' and 'q', then combining the results with OR.
Truth tables are a method to visually represent these logical operations. They show how the output of an operation changes based on different combinations of input values.
Through practicing logical operations, students can gain a better grasp of the mechanics of logic and learn to anticipate the outcomes of complex statements.

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Most popular questions from this chapter

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=(p \rightarrow q) \rightarrow r, Q=p \rightarrow(q \rightarrow r) $$

Suppose that \(P\) is a propositional function with domain of discourse \(\left\\{d_{1}, \ldots, d_{n}\right\\} \times\left\\{d_{1}, \ldots, d_{n}\right\\} .\) Write pseudocode that determines whether $$ \exists x \exists y P(x, y) $$ is true or false.

Verify the second of De Morgan's laws, \(\neg(p \wedge q) \equiv \neg p \vee \neg q\).

List the members of \(\mathcal{P}(\\{a, b, c, d\\}) .\) Which are proper subsets of \(\\{a, b, c, d\\} ?\)

For each pair of propositions \(P\) and \(Q\) . State whether or not \(P \equiv Q\). $$ P=(s \rightarrow(p \wedge \neg r)) \wedge((p \rightarrow(r \vee q)) \wedge s), Q=p \vee t $$
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