Problem 54
Question
Let \(p\) and \(q\) represent the following simple statements: p: Romeo loves Juliet. q: Juliet loves Romeo. Write each symbolic statement in words. \(\sim(p \vee q)\)
Step-by-Step Solution
Verified Answer
The statement \(\sim(p \vee q)\) translates to 'Neither Romeo loves Juliet nor Juliet loves Romeo' in English.
1Step 1: Understanding the symbols
The symbols are explained as follows: 'p' stands for the statement 'Romeo loves Juliet', 'q' stands for the statement 'Juliet loves Romeo', '\(\sim\)' is the negation operator (negating the following statement), and '\(\vee\)' stands for logical OR (it means 'either... or... both')
2Step 2: Understanding the Logical OR
The logical OR operation, \(\vee\), means that at least one of the combined statements is true. So, in this case, \(p \vee q\) would translate to 'either Romeo loves Juliet or Juliet loves Romeo or both loves each other'.
3Step 3: Applying the Negation
The negation operator, \(\sim\), negates the following statement. So, \(\sim(p \vee q)\) results in 'It is not the case that either Romeo loves Juliet or Juliet loves Romeo or both loves each other'.
Key Concepts
Understanding the Negation OperatorExploring the Logical ORWorking with Symbolic Statements
Understanding the Negation Operator
The negation operator, denoted by the symbol \( \sim \), is used to reverse or negate the truth value of a statement. When you apply this operator to a statement, it means "not" or "it is not the case that" the statement is true.
Here's how it works: If \( p \) represents the statement "Romeo loves Juliet," then \( \sim p \) represents "Romeo does not love Juliet." The negation essentially flips the truth value. If the original statement is true, the negated statement is false, and vice versa.
Here's how it works: If \( p \) represents the statement "Romeo loves Juliet," then \( \sim p \) represents "Romeo does not love Juliet." The negation essentially flips the truth value. If the original statement is true, the negated statement is false, and vice versa.
Exploring the Logical OR
The logical OR, represented by \( \vee \), is a fundamental operation in logic. It states that at least one of the connected statements must be true. It can represent either or both conditions being true.
For instance, if \( p \) is "Romeo loves Juliet" and \( q \) is "Juliet loves Romeo," then \( p \vee q \) translates to "either Romeo loves Juliet, or Juliet loves Romeo, or both." This allows for multiple possibilities: only \( p \) is true, only \( q \) is true, or both are true.
For instance, if \( p \) is "Romeo loves Juliet" and \( q \) is "Juliet loves Romeo," then \( p \vee q \) translates to "either Romeo loves Juliet, or Juliet loves Romeo, or both." This allows for multiple possibilities: only \( p \) is true, only \( q \) is true, or both are true.
Working with Symbolic Statements
Symbolic statements use symbols to represent logical expressions, making complex relationships easier to visualize and manipulate.
In the original example, the symbolic statement \( \sim(p \vee q) \) uses symbols to communicate a complex idea succinctly. Here's how you translate it:
In the original example, the symbolic statement \( \sim(p \vee q) \) uses symbols to communicate a complex idea succinctly. Here's how you translate it:
- \( p \) represents "Romeo loves Juliet."
- \( q \) represents "Juliet loves Romeo."
- \( \vee \) means "or."
- \( \sim \) negates the entire expression \( (p \vee q) \).
Other exercises in this chapter
Problem 54
Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \([(p \rightarrow q) \wedge(q \rightarrow r)] \rightarr
View solution Problem 54
Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(p \vee(q \wedge r)\)
View solution Problem 55
Use a truth table to determine whether each statement is a tautology, a self- contradiction, or neither. \([(q \rightarrow r) \wedge(r \longrightarrow \sim p)]
View solution Problem 55
Determine the truth value for each statement when \(p\) is false, \(q\) is true, and \(r\) is false. \(\sim p \vee(q \wedge \sim r)\)
View solution