Question:
Let \( p, q, r \) be three simple statements. Then \( \sim(p \lor q) \lor \sim(p \lor r) \equiv \)
Let \( p, q, r \) be three simple statements. Then \( \sim(p \lor q) \lor \sim(p \lor r) \equiv \)
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When simplifying logical expressions, think of $\land$ like multiplication and $\lor$ like addition. Factoring out common terms (Distributive Law) is often the fastest way to reach the simplified form.
Updated On: May 2, 2026
- $(\sim p) \land (\sim q \lor \sim r)$
- $(\sim p) \land (q \lor r)$
- $p \land (q \lor r)$
- $p \lor (q \land r)$
- $(p \lor q) \land r$
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The Correct Option is A
Solution and Explanation
Concept:
This problem involves De Morgan's Laws and the Distributive Law of logical equivalence.
• De Morgan's Law: $\sim(a \lor b) \equiv \sim a \land \sim b$.
• Distributive Law: $(a \land b) \lor (a \land c) \equiv a \land (b \lor c)$.
Step 1: Apply De Morgan's Law to each component.
The given expression is $\sim(p \lor q) \lor \sim(p \lor r)$. Applying $\sim(A \lor B) \equiv \sim A \land \sim B$: \[ \sim(p \lor q) \equiv \sim p \land \sim q \] \[ \sim(p \lor r) \equiv \sim p \land \sim r \] So the expression becomes: \[ (\sim p \land \sim q) \lor (\sim p \land \sim r) \]
Step 2: Apply the Distributive Law.
Notice that $\sim p$ is a common factor in both terms joined by an OR ($\lor$): \[ (\sim p \land \sim q) \lor (\sim p \land \sim r) \equiv \sim p \land (\sim q \lor \sim r) \]
• De Morgan's Law: $\sim(a \lor b) \equiv \sim a \land \sim b$.
• Distributive Law: $(a \land b) \lor (a \land c) \equiv a \land (b \lor c)$.
Step 1: Apply De Morgan's Law to each component.
The given expression is $\sim(p \lor q) \lor \sim(p \lor r)$. Applying $\sim(A \lor B) \equiv \sim A \land \sim B$: \[ \sim(p \lor q) \equiv \sim p \land \sim q \] \[ \sim(p \lor r) \equiv \sim p \land \sim r \] So the expression becomes: \[ (\sim p \land \sim q) \lor (\sim p \land \sim r) \]
Step 2: Apply the Distributive Law.
Notice that $\sim p$ is a common factor in both terms joined by an OR ($\lor$): \[ (\sim p \land \sim q) \lor (\sim p \land \sim r) \equiv \sim p \land (\sim q \lor \sim r) \]
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