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It looks like you've uploaded an image of a truth table for a logical problem involving the statements \( p \), \( q \), and \( r \). To help you, I'll complete the table step by step for the compound statements shown.

I’ll start by interpreting the given logic:

- \( p \), \( q \), and \( r \) are the individual statements.
- The first column contains their possible truth values, which we'll combine in all 8 combinations of true (T) and false (F).
- The table asks us to evaluate different compound statements such as negations and logical connectives (AND, OR, implication, etc.).

Let me calculate the values for the entire truth table for you. pandas as pd

# Truth values for p, q, r
truth_values = ['T', 'F']
# Generate all possible combinations of truth values for p, q, and r
truth_combinations = [(p, q, r) for p in truth_values for q in truth_values for r in truth_values]

# Function to evaluate logical expressions
def evaluate_expression(p, q, r):
    # Negations
    not_p = 'F' if p == 'T' else 'T'
    not_q = 'F' if q == 'T' else 'T'
    not_r = 'F' if r == 'T' else 'T'
    
    # Compound statements
    p_and_q = 'T' if p == 'T' and q == 'T' else 'F'
    p_or_r = 'T' if p == 'T' or r == 'T' else 'F'
    p_implies_r = 'T' if p == 'F' or r == 'T' else 'F'
    not_p_and_q = 'T' if not_p == 'T' and q == 'T' else 'F'
    r_or_not_q = 'T' if r == 'T' or not_q == 'T' else 'F'
    not_r_implies_not_q = 'T' if not_r == 'F' or not_q == 'T' else 'F'
    
    return (not_p, not_q, not_r, p_and_q, p_or_r, p_implies_r, not_p_and_q, r_or_not_q, not_r_implies_not_q)

# Apply the evaluation to each combination
results = [evaluate_expression(p, q, r) for p, q, r in truth_combinations]

# Create a DataFrame for the result
df = pd.DataFrame(results, columns=[
    "¬p", "¬q", "¬r", "p∧q", "p∨r", "p⇒r", "¬p∧q", "r∨¬q", "¬r⇒¬q"
])

# Add the original columns for p, q, r
df.insert(0, 'p', [p for p, _, _ in truth_combinations])
df.insert(1, 'q', [q for _, q, _ in truth_combinations])
df.insert(2, 'r', [r for _, _, r in truth_combinations])

dfHere is the completed truth table for the given compound statements:

\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
p & q & r & \neg p & \neg q & \neg r & p \land q & p \lor r & p \Rightarrow r & \neg p \land q & r \lor \neg q & \neg r \Rightarrow \neg q \\
\hline
T & T & T & F & F & F & T & T & T & F & T & T \\
T & T & F & F & F & T & T & T & F & F & F & F \\
T & F & T & F & T & F & F & T & T & F & T & T \\
T & F & F & F & T & T & F & T & F & F & T & T \\
F & T & T & T & F & F & F & T & T & T & T & T \\
F & T & F & T & F & T & F & F & T & T & F & F \\
F & F & T & T & T & F & F & T & T & F & T & T \\
F & F & F & T & T & T & F & F & T & F & T & T \\
\hline
\end{array}
\]

This table lists all the truth values for the given propositions and their logical combinations.

Would you like me to explain any part of the table, or do you have any further questions on logical operations or truth tables?

Here are 5 related questions to deepen the understanding:
1. What are the basic logical operators used in the table?
2. How do you evaluate a logical implication (like \( p \Rightarrow q \)) in a truth table?
3. What is the difference between conjunction (\( \land \)) and disjunction (\( \lor \))?
4. How does negation (\( \neg \)) affect the truth values of a statement?
5. Can you simplify compound statements using logical identities based on this table?

**Tip**: When constructing truth tables, ensure you calculate the intermediate logical steps (like negations and conjunctions) before evaluating the final compound statements.

Let p, q and r be three statements. Construct the truth table for the compound statements by completing the table below.

Learn how to construct a truth table for logical statements involving negation, conjunction, disjunction, and implication. This step-by-step guide covers all possible truth values for the given statements p, q, and r, suitable for high school students studying logic and truth tables.

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