Questions: Construct a truth table for the argument. Use "T" for true and "F" for false. p q r p ↔ q q ↔ r p ∧ q (p ↔ q) ∧ (q ↔ r) → p ∧ q -------------------------------------------------------------- T T T T T F T F T T F F F T T F T F F F T F F F

Construct a truth table for the argument. Use "T" for true and "F" for false. p q r p ↔ q q ↔ r p ∧ q (p ↔ q) ∧ (q ↔ r) → p ∧ q -------------------------------------------------------------- T T T T T F T F T T F F F T T F T F F F T F F F
Transcript text: Construct a truth table for the argument. Use "T" for true and "F" for false. \begin{tabular}{|c|c|c|c|c|c|c|} \hline$p$ & $q$ & r & $p \leftrightarrow q$ & $q \leftrightarrow r$ & $p \wedge q$ & $(p \leftrightarrow q) \wedge(q \leftrightarrow r) \rightarrow p \wedge q$ \\ \hline T & T & T & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline T & T & F & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline T & F & T & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline T & F & F & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline F & T & T & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline F & T & F & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline F & F & T & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline F & F & F & $\square$ & $\square$ & $\square$ & $\square$ \\ \hline \end{tabular}
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Solution

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Solution Steps

To construct a truth table for the given logical argument, we need to evaluate each expression for all possible truth values of the variables \( p \), \( q \), and \( r \). We will compute the truth values for each column in the table: \( p \leftrightarrow q \), \( q \leftrightarrow r \), \( p \wedge q \), and finally the compound statement \((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\).

Step 1: Define the Logical Operations

To construct the truth table, we define the logical operations needed: biconditional (\(\leftrightarrow\)), conjunction (\(\wedge\)), and implication (\(\rightarrow\)). These operations will help us evaluate the truth values of the expressions in the table.

Step 2: Generate All Possible Combinations of Truth Values

We consider all possible combinations of truth values for the variables \( p \), \( q \), and \( r \). There are \( 2^3 = 8 \) combinations, as each variable can be either true (T) or false (F).

Step 3: Evaluate Each Expression

For each combination of truth values, we evaluate the following expressions:

  • \( p \leftrightarrow q \)
  • \( q \leftrightarrow r \)
  • \( p \wedge q \)
  • \((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\)
Step 4: Construct the Truth Table

Using the evaluated expressions, we construct the truth table:

\[ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & r & p \leftrightarrow q & q \leftrightarrow r & p \wedge q & (p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q \\ \hline T & T & T & T & T & T & T \\ T & T & F & T & F & T & T \\ T & F & T & F & F & F & T \\ T & F & F & F & T & F & T \\ F & T & T & F & T & F & T \\ F & T & F & F & F & F & T \\ F & F & T & T & F & F & T \\ F & F & F & T & T & F & F \\ \hline \end{array} \]

Final Answer

The truth table for the given logical argument is constructed as shown above. The final column, which represents the compound statement \((p \leftrightarrow q) \wedge (q \leftrightarrow r) \rightarrow p \wedge q\), is true for all combinations except when \( p = F \), \( q = F \), and \( r = F \).

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