Step 1: Understanding the Concept:
We need to evaluate the truth value of the given logical statement pattern for all possible combinations of truth values for \( p \) and \( q \).
We can do this either by constructing a full truth table or by using laws of logic to simplify the expression.
Step 2: Key Formula or Approach:
We will use logical equivalences to simplify the two main parts of the expression:
Part 1: \( p \rightarrow (q \land \sim p) \)
Part 2: \( (p \lor \sim q) \land p \)
Recall the implication equivalence: \( A \rightarrow B \equiv \sim A \lor B \).
Recall the absorption law: \( A \land (A \lor B) \equiv A \).
Step 3: Detailed Explanation:
Let the given expression be \( E = E_1 \lor E_2 \), where \( E_1 = p \rightarrow (q \land \sim p) \) and \( E_2 = (p \lor \sim q) \land p \).
Let's simplify \( E_1 \):
\[ E_1 \equiv \sim p \lor (q \land \sim p) \]
Using distributive law:
\[ E_1 \equiv (\sim p \lor q) \land (\sim p \lor \sim p) \]
\[ E_1 \equiv (\sim p \lor q) \land \sim p \]
Using commutative and absorption laws (or just basic logic: if \( \sim p \) is true, the whole expression is true; if \( \sim p \) is false, the expression is false):
\[ E_1 \equiv \sim p \land (\sim p \lor q) \equiv \sim p \]
Let's simplify \( E_2 \):
\[ E_2 = (p \lor \sim q) \land p \]
By commutative law:
\[ E_2 = p \land (p \lor \sim q) \]
By the absorption law, this directly simplifies to \( p \).
\[ E_2 \equiv p \]
Now, combine \( E_1 \) and \( E_2 \) back into the original expression \( E \):
\[ E = E_1 \lor E_2 \equiv \sim p \lor p \]
The statement \( \sim p \lor p \) is a tautology, meaning it is always True regardless of the truth values of \( p \) and \( q \).
Therefore, the last column of the truth table will consist entirely of 'T's.
The sequence of truth values for the standard order (TT, TF, FT, FF) is T, T, T, T.
Step 4: Final Answer:
The last column is TTTT.