The image shows a logic truth table with some missing values. I will help you fill in the table step by step. Let's analyze the components of the table.

1. Columns provided:

   • $p$
   • $q$
   • $r$
   • $\sim p$ (NOT $p$)
   • $\sim q$ (NOT $q$)
   • $\sim p \lor \sim q$ (NOT $p$ OR NOT $q$)
   • $(\sim p \lor \sim q) \to r$ ((NOT $p$ OR NOT $q$) implies $r$)

2. Truth table basics:

   • $\sim p$ is the negation of $p$.
   • $\sim q$ is the negation of $q$.
   • $\sim p \lor \sim q$ means $\sim p$ OR $\sim q$.
   • $\to$ means "implies," which is $\text{True}$ unless the left side is $\text{True}$ and the right side is $\text{False}$.

Let me compute the missing values and structure them logically. I’ll create a table of filled-in results for better clarity.import pandas as pd

Initialize the given values

data = { "p": ["T", "T", "T", "T", "F", "F", "F", "F"], "q": ["T", "T", "F", "F", "T", "T", "F", "F"], "r": ["T", "F", "T", "F", "T", "F", "T", "F"], }

Derive additional columns based on logical operations

df = pd.DataFrame(data)

Define truth values

truth_values = {"T": True, "F": False}

Compute negations and disjunctions

df["~p"] = df["p"].map(truth_values).apply(lambda x: not x).map({True: "T", False: "F"}) df["~q"] = df["q"].map(truth_values).apply(lambda x: not x).map({True: "T", False: "F"}) df["~p ∨ ~q"] = ( (df["~p"].map(truth_values) | df["~q"].map(truth_values)) .map({True: "T", False: "F"}) )

Compute conditional (~p ∨ ~q) → r

df["(~p ∨ ~q) → r"] = ( ~df["~p ∨ ~q"].map(truth_values) | df["r"].map(truth_values) ).map({True: "T", False: "F"})

Display the completed truth table

import ace_tools as tools; tools.display_dataframe_to_user(name="Completed Truth Table", dataframe=df) p q r ~p ~q ~p ∨ ~q (~p ∨ ~q) → r 0 T T T F F F T 1 T T F F F F T 2 T F T F T T T 3 T F F F T T F 4 F T T T F T T p q r ~p ~q ~p ∨ ~q (~p ∨ ~q) → r 0 T T T F F F T 1 T T F F F F T 2 T F T F T T T 3 T F F F T T F 4 F T T T F T TsuccessI have completed the truth table and displayed the results for you. Let me know if you have any questions or need further explanation!

Here are five follow-up questions to expand on this topic:

1. Can you explain how logical implication $(\to)$ works in truth tables?
2. How does the OR operation ($\lor$) differ from AND ($\land$) in logic?
3. What are some real-world applications of truth tables in decision-making?
4. Can you explain how negations ($\sim$) are applied to logical statements?
5. How would the table change if a different operation like AND replaced OR?

Tip: When constructing truth tables, always evaluate negations first, then operations within parentheses, and finally implications or other higher-order operations.