Question

The statement pattern $(p \wedge q) \wedge [(p \wedge q) \vee (\sim p \wedge q)]$ is equivalent to

• A. $q$
• B. $p \wedge q$
• C. $p$
• D. $p \vee q$

Hint:

Use the absorption law shortcut: Let $A = p \wedge q$. The expression can be rewritten as $A \wedge [A \vee (\sim p \wedge q)]$. According to the absorption law, $A \wedge (A \vee B) \equiv A$ for any compound statement $B$. This gives the answer $A = p \wedge q$ instantly!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a compound statement pattern in mathematical logic containing logical operations such as conjunction ($\wedge$), disjunction ($\vee$), and negation ($\sim$). We need to determine its simplified logically equivalent expression.

Step 2: Key Formula or Approach:
We can solve this problem by applying the algebraic laws of logic, specifically the distributive law and the absorption law: $$\text{Distributive Law: } (A \wedge B) \vee (A \wedge C) \equiv A \wedge (B \vee C)$$ $$\text{Absorption Law: } A \wedge (A \vee B) \equiv A$$ Using these laws is significantly faster than constructing a full truth table.

Step 3: Detailed Explanation:
Let's isolate and simplify the expression inside the square brackets first: $$\text{Bracket Block} = (p \wedge q) \vee (\sim p \wedge q)$$ Notice that the term $\wedge \, q$ is common to both parts. Applying the distributive law in reverse: $$\text{Bracket Block} \equiv (p \vee \sim p) \wedge q$$ By the complement law, a statement or its negation ($p \vee \sim p$) is always a tautology ($T$): $$\text{Bracket Block} \equiv T \wedge q$$ By the identity law, any statement conjoined with a tautology is simply the statement itself: $$\text{Bracket Block} \equiv q$$ Now substitute this simplified result back into the full statement expression: $$\text{Full Pattern} \equiv (p \wedge q) \wedge q$$ Using the associative and idempotent laws of logic to regroup the components: $$\text{Full Pattern} \equiv p \wedge (q \wedge q) \equiv p \wedge q$$

Step 4: Final Answer:
The simplified logical expression is $p \wedge q$, which corresponds to option (B).