Problem 30
Question
Give the truth value of each proposition, using the given information. \(p \vee q,\) where \(\sim p\) is not true.
Step-by-Step Solution
Verified Answer
Since \(p\) is true and the logical disjunction \(p \vee q\) is true if either or both \(p\) and \(q\) are true, the truth value of \(p \vee q\) is true.
1Step 1: Identify the Given Information
We are given the following:
1. \(p \vee q\) is the compound proposition.
2. \(\sim p\) is not true.
From this, we can immediately conclude that \(p\) is true.
2Step 2: Define the Logical Disjunction
The logical disjunction \(p \vee q\) is true if either or both \(p\) and \(q\) are true.
3Step 3: Apply the Definition of the Logical Disjunction
Since we already know that \(p\) is true, we can use the definition of the logical disjunction to find the truth value of the compound proposition \(p \vee q\):
We have \(p\) = true, and we are looking for \(p \vee q\). By the definition of logical disjunction, \(p \vee q\) will be true if at least one of \(p\) or \(q\) is true. As we know that \(p\) is true, it is enough to conclude that the compound proposition \(p \vee q\) is true. Therefore, the truth value of \(p \vee q\) is true.
Key Concepts
Truth ValueCompound PropositionLogical Operators
Truth Value
To understand logical disjunction, it is essential to start with the concept of truth value. A truth value refers to the attribute that any proposition (a statement that is either true or false) possesses, which reflects its truthfulness. In the context of logic, propositions are typically assigned a truth value of either 'true' or 'false'.
Considering the exercise, when we are presented with the statement that \(\sim p\) is not true, we can immediately infer that the truth value of \(p\) is 'true'. This understanding is fundamental when evaluating the truth of compound propositions, as the determination rests on the truth values of their constituent statements.
Considering the exercise, when we are presented with the statement that \(\sim p\) is not true, we can immediately infer that the truth value of \(p\) is 'true'. This understanding is fundamental when evaluating the truth of compound propositions, as the determination rests on the truth values of their constituent statements.
Compound Proposition
A compound proposition is formed by combining two or more simple propositions using logical operators. They are structured to express more complex conditions that can't be conveyed by single propositions. It's crucial for students to recognize that the truth value of a compound proposition depends on its structure, as well as the truth values of its constituent propositions.
In the given exercise, \(p \vee q\) is a compound proposition. It combines the simple propositions \(p\) and \(q\) using the logical operator 'logical disjunction'. The compound does not rely solely on the truth of \(p\) or \(q\), but rather on the application of the rules of the logical operator used to combine them.
In the given exercise, \(p \vee q\) is a compound proposition. It combines the simple propositions \(p\) and \(q\) using the logical operator 'logical disjunction'. The compound does not rely solely on the truth of \(p\) or \(q\), but rather on the application of the rules of the logical operator used to combine them.
Logical Operators
Logical operators, also known as logical connectives, are symbols or words used to connect propositions in such a way that they express a relationship between them. They dictate how the truth values of simple propositions influence the truth value of a compound proposition. Common logical operators include AND (\(\land\)), OR (\(\vee\)), NOT (\(\sim\)), and IF...THEN (\(\rightarrow\)).
In our case, the logical operator of interest is 'OR' which is represented by 'logical disjunction' (\(\vee\)). The disjunction of \(p\) or \(q\), \(p \vee q\), is true if at least one of the propositions \(p\) or \(q\) is true. With the knowledge that \(p\) is true in the exercise, we can directly apply the logical disjunction to conclude that the compound proposition \(p \vee q\) is also true, regardless of the truth value of \(q\). This illustrates the importance of logical operators for determining the overall truth of compound propositions.
In our case, the logical operator of interest is 'OR' which is represented by 'logical disjunction' (\(\vee\)). The disjunction of \(p\) or \(q\), \(p \vee q\), is true if at least one of the propositions \(p\) or \(q\) is true. With the knowledge that \(p\) is true in the exercise, we can directly apply the logical disjunction to conclude that the compound proposition \(p \vee q\) is also true, regardless of the truth value of \(q\). This illustrates the importance of logical operators for determining the overall truth of compound propositions.
Other exercises in this chapter
Problem 30
Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \wedge t \equiv p$$
View solution Problem 30
Prove by the existence method. There are integers \(x\) such that \(|x|=x\)
View solution Problem 30
Prove by the existence method. There are integers \(x\) such that \(|x|=x\).
View solution Problem 31
Mark each sentence as true or false, where \(p, q,\) and \(r\) are arbitrary statements, \(t\) a tautology, and \(f\) a contradiction. $$p \vee f \equiv p$$
View solution