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Complete the truth table for the given statement by filling in the required columns. -pv-q Pq Р TT T F FT FF -P -q -pv-q

Complete the truth table for the given statement by filling in the required columns. -pv-q Pq Р TT T F FT FF -P -q -pv-q

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Truth Table Exercise This educational page provides guidance on completing a truth table for the logical expression \(\sim p \vee \sim q\). Below is an explanation and breakdown of how to complete the truth table. #### Instructions: 1. **Table 1:** Begin by filling out the empty columns in the given truth table for the expression \(\sim p \vee \sim q\). 2. **Table 1 Details:** | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-----|-----|------------|------------|------------------------| | T | T | | | | | T | F | | | | | F | T | | | | | F | F | | | | - **p and q:** These columns list all possible truth values for the propositions p and q, True (T) or False (F). - **\(\sim p\) and \(\sim q\):** Calculate the negation of p and q respectively. - **\(\sim p \vee \sim q\):** Use the outcomes from \(\sim p\) and \(\sim q\) to evaluate the disjunction. #### Practicing Filling Out the Table 3. **Interactive Table Completion:** In the interactive section, select the correct logical value for each cell based on the negation and disjunction calculations. 4. **Table 2 (Interactive Practice):** | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-----|-----|------------|------------|------------------------| | T | T | ⬇️ | ⬇️ | ⬇️ | | T | F | ⬇️ | ⬇️ | ⬇️ | | F | T | ⬇️ | ⬇️ | ⬇️ | | F | F | ⬇️ | ⬇️ | ⬇️
Transcribed Image Text:### Truth Table Exercise This educational page provides guidance on completing a truth table for the logical expression \(\sim p \vee \sim q\). Below is an explanation and breakdown of how to complete the truth table. #### Instructions: 1. **Table 1:** Begin by filling out the empty columns in the given truth table for the expression \(\sim p \vee \sim q\). 2. **Table 1 Details:** | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-----|-----|------------|------------|------------------------| | T | T | | | | | T | F | | | | | F | T | | | | | F | F | | | | - **p and q:** These columns list all possible truth values for the propositions p and q, True (T) or False (F). - **\(\sim p\) and \(\sim q\):** Calculate the negation of p and q respectively. - **\(\sim p \vee \sim q\):** Use the outcomes from \(\sim p\) and \(\sim q\) to evaluate the disjunction. #### Practicing Filling Out the Table 3. **Interactive Table Completion:** In the interactive section, select the correct logical value for each cell based on the negation and disjunction calculations. 4. **Table 2 (Interactive Practice):** | p | q | \(\sim p\) | \(\sim q\) | \(\sim p \vee \sim q\) | |-----|-----|------------|------------|------------------------| | T | T | ⬇️ | ⬇️ | ⬇️ | | T | F | ⬇️ | ⬇️ | ⬇️ | | F | T | ⬇️ | ⬇️ | ⬇️ | | F | F | ⬇️ | ⬇️ | ⬇️
**Refer to the following statement to answer parts (a) through (c) below.** The mathematician is intelligent and an overachiever, or not an overachiever. --- **Truth Table:** | p | q | p ∧ q | ~q | (p ∧ q) ∨ ~q | |-----|-----|-------|-----|---------------| | T | T | | | | | T | F | | | | | F | T | | | | | F | F | | | | --- **c.** Use the truth table to indicate one set of conditions that make the compound statement true, or state that no such conditions exist. Choose the correct answer below. - A. The statement is true when p is true and q is false. - B. The statement is true for all conditions. - C. The statement is true when p is true or q is false. - D. There are no conditions for which the statement is true. --- **Explanation of Diagram:** The table provided is a truth table, which helps evaluate the truth values of a compound logical statement based on various combinations of truth values for its components. The columns titled "p" and "q" represent two statements, where "T" stands for true, and "F" stands for false. - "p ∧ q" indicates a logical conjunction (both p and q need to be true for this to be true). - "~q" represents the negation of q (this is true when q is false). - "(p ∧ q) ∨ ~q" indicates a logical disjunction, meaning the statement is true if either p ∧ q is true, or ~q is true (or both). The task is to fill in the truth table and determine under what conditions the compound statement is true.
Transcribed Image Text:**Refer to the following statement to answer parts (a) through (c) below.** The mathematician is intelligent and an overachiever, or not an overachiever. --- **Truth Table:** | p | q | p ∧ q | ~q | (p ∧ q) ∨ ~q | |-----|-----|-------|-----|---------------| | T | T | | | | | T | F | | | | | F | T | | | | | F | F | | | | --- **c.** Use the truth table to indicate one set of conditions that make the compound statement true, or state that no such conditions exist. Choose the correct answer below. - A. The statement is true when p is true and q is false. - B. The statement is true for all conditions. - C. The statement is true when p is true or q is false. - D. There are no conditions for which the statement is true. --- **Explanation of Diagram:** The table provided is a truth table, which helps evaluate the truth values of a compound logical statement based on various combinations of truth values for its components. The columns titled "p" and "q" represent two statements, where "T" stands for true, and "F" stands for false. - "p ∧ q" indicates a logical conjunction (both p and q need to be true for this to be true). - "~q" represents the negation of q (this is true when q is false). - "(p ∧ q) ∨ ~q" indicates a logical disjunction, meaning the statement is true if either p ∧ q is true, or ~q is true (or both). The task is to fill in the truth table and determine under what conditions the compound statement is true.
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