Let p be "an animal is a dog" and let q be "an animal is a golder If an animal is not a dog, then it is not a golden retriever. 0 P false Op true 04P true

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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### Logical Statements and Their Truth Values

#### Instructions
Write each statement using symbols. Then, decide whether each statement is true or false.

#### Symbols Definition
Let \( p \) be "an animal is a dog" and let \( q \) be "an animal is a golden retriever".

#### Statement Analysis
**If an animal is not a dog, then it is not a golden retriever.**

#### Options
1. \( \sim q \rightarrow p \), false
2. \( \sim p \rightarrow \sim q \), false
3. \( \sim p \rightarrow \sim q \), true
4. \( \sim q \rightarrow \sim p \), true

#### Explanation
- A conditional statement in logic takes the form \( A \rightarrow B \) which translates to "If A, then B".
- The negation of a statement \( p \) is denoted as \( \sim p \).
- The problem asks us to express the given statement in symbolic form and then evaluate its truth value based on logical relations.

Given the statement "If an animal is not a dog ( \( \sim p \) ), then it is not a golden retriever ( \( \sim q \) )":

- Statement \( \sim p \rightarrow \sim q \) correctly represents this.
- Therefore, option 3 \( \sim p \rightarrow \sim q \), true, is the correct answer.
Transcribed Image Text:### Logical Statements and Their Truth Values #### Instructions Write each statement using symbols. Then, decide whether each statement is true or false. #### Symbols Definition Let \( p \) be "an animal is a dog" and let \( q \) be "an animal is a golden retriever". #### Statement Analysis **If an animal is not a dog, then it is not a golden retriever.** #### Options 1. \( \sim q \rightarrow p \), false 2. \( \sim p \rightarrow \sim q \), false 3. \( \sim p \rightarrow \sim q \), true 4. \( \sim q \rightarrow \sim p \), true #### Explanation - A conditional statement in logic takes the form \( A \rightarrow B \) which translates to "If A, then B". - The negation of a statement \( p \) is denoted as \( \sim p \). - The problem asks us to express the given statement in symbolic form and then evaluate its truth value based on logical relations. Given the statement "If an animal is not a dog ( \( \sim p \) ), then it is not a golden retriever ( \( \sim q \) )": - Statement \( \sim p \rightarrow \sim q \) correctly represents this. - Therefore, option 3 \( \sim p \rightarrow \sim q \), true, is the correct answer.
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