Question:
The expression $[(p \wedge \sim q) \vee q] \vee (\sim p \wedge q)$ is equivalent to
The expression $[(p \wedge \sim q) \vee q] \vee (\sim p \wedge q)$ is equivalent to
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When resolving logical equivalences in an exam, constructing a quick truth table or substituting truth values can save time. For instance, if you set $p = \text{True}$ and $q = \text{False}$, the expression evaluates to $[(\text{True} \wedge \text{True}) \vee \text{False}] \vee (\text{False} \wedge \text{False}) = \text{True}$. Checking the options, $p \vee q$ is $\text{True}$, which immediately eliminates options (B) and (C).
Updated On: Jun 18, 2026
- $p \vee q$
- $p \wedge q$
- $p \rightarrow q$
- $p \leftrightarrow q$
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The Correct Option is A
Solution and Explanation
Step 1: Understanding the Question:
The question asks us to simplify a composite mathematical logic expression containing the propositions $p$ and $q$ along with the operators $\wedge$ (AND), $\vee$ (OR), and $\sim$ (NOT) to find its logical equivalent.
Step 2: Key Formula or Approach:
We can use the algebraic laws of logic to simplify the expression systematically: Distributive Law: $(A \wedge B) \vee C \equiv (A \vee C) \wedge (B \vee C)$ Complement Law: $A \vee \sim A \equiv \text{True}$ Identity Law: $A \wedge \text{True} \equiv A$
Step 3: Detailed Explanation:
Let's first simplify the inner bracket expression: $[(p \wedge \sim q) \vee q]$.
Applying the Distributive Law inside the bracket: $$[(p \vee q) \wedge (\sim q \vee q)]$$ By the Complement Law, we know that $\sim q \vee q \equiv \text{True}$: $$[(p \vee q) \wedge \text{True}]$$ Using the Identity Law, any statement ANDed with True is the statement itself: $$p \vee q$$ Now substitute this back into the original expression: $$(p \vee q) \vee (\sim p \wedge q)$$ Applying the Distributive Law again to expand the expression: $$[(p \vee q) \vee \sim p] \wedge [(p \vee q) \vee q]$$ Using the Commutative and Associative Laws on both parts: First part: $(p \vee \sim p) \vee q \equiv \text{True} \vee q \equiv \text{True}$ Second part: $p \vee (q \vee q) \equiv p \vee q$ Combining both simplified parts back together with the $\wedge$ operator: $$\text{True} \wedge (p \vee q) \equiv p \vee q$$
Step 4: Final Answer:
The simplified expression is logically equivalent to $p \vee q$, which corresponds to option (A).
The question asks us to simplify a composite mathematical logic expression containing the propositions $p$ and $q$ along with the operators $\wedge$ (AND), $\vee$ (OR), and $\sim$ (NOT) to find its logical equivalent.
Step 2: Key Formula or Approach:
We can use the algebraic laws of logic to simplify the expression systematically: Distributive Law: $(A \wedge B) \vee C \equiv (A \vee C) \wedge (B \vee C)$ Complement Law: $A \vee \sim A \equiv \text{True}$ Identity Law: $A \wedge \text{True} \equiv A$
Step 3: Detailed Explanation:
Let's first simplify the inner bracket expression: $[(p \wedge \sim q) \vee q]$.
Applying the Distributive Law inside the bracket: $$[(p \vee q) \wedge (\sim q \vee q)]$$ By the Complement Law, we know that $\sim q \vee q \equiv \text{True}$: $$[(p \vee q) \wedge \text{True}]$$ Using the Identity Law, any statement ANDed with True is the statement itself: $$p \vee q$$ Now substitute this back into the original expression: $$(p \vee q) \vee (\sim p \wedge q)$$ Applying the Distributive Law again to expand the expression: $$[(p \vee q) \vee \sim p] \wedge [(p \vee q) \vee q]$$ Using the Commutative and Associative Laws on both parts: First part: $(p \vee \sim p) \vee q \equiv \text{True} \vee q \equiv \text{True}$ Second part: $p \vee (q \vee q) \equiv p \vee q$ Combining both simplified parts back together with the $\wedge$ operator: $$\text{True} \wedge (p \vee q) \equiv p \vee q$$
Step 4: Final Answer:
The simplified expression is logically equivalent to $p \vee q$, which corresponds to option (A).
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