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Use DeMorgan's Laws to write an equivalent statement in symbolic form. Select the correct choice. \( \sim(\sim p \vee q) \) \( O p \vee q \) \( O p \wedge q \) \( \sim p \vee q \) \( \sim p \wedge q \) \( p p \vee \sim q \) \( \sim p \wedge \sim q \) \( \sim p \vee \sim q \) \( \sim p \wedge \sim q \)

Ask by Turner Boyd. in the United States
Mar 10,2025

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Answer

The equivalent statement is \( p \wedge \sim q \).

Solution

To apply DeMorgan's Laws to the expression \( \sim(\sim p \vee q) \), we need to recall the laws themselves: 1. \( \sim (A \vee B) \equiv \sim A \wedge \sim B \) 2. \( \sim (A \wedge B) \equiv \sim A \vee \sim B \) In our case, we have \( A = \sim p \) and \( B = q \). Therefore, we can apply the first law: \[ \sim(\sim p \vee q) \equiv \sim(\sim p) \wedge \sim(q) \] Now, simplifying \( \sim(\sim p) \) gives us \( p \). Thus, we have: \[ \sim(\sim p \vee q) \equiv p \wedge \sim q \] Now, let's look at the options provided: 1. \( O p \vee q \) 2. \( O p \wedge q \) 3. \( \sim p \vee q \) 4. \( \sim p \wedge q \) 5. \( p p \vee \sim q \) 6. \( \sim p \wedge \sim q \) 7. \( \sim p \vee \sim q \) 8. \( \sim p \wedge \sim q \) None of the options directly match \( p \wedge \sim q \). However, it seems there might be a typographical error in the options, particularly with "O" and "p p". The correct equivalent statement in symbolic form, based on DeMorgan's Laws, is: \[ p \wedge \sim q \] If you have the option that closely resembles this, please let me know!

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Bonus Knowledge

To apply DeMorgan's Laws to the statement \( \sim(\sim p \vee q) \), we transform it into \( \sim(\sim p) \wedge \sim(q) \). This simplifies to \( p \wedge \sim q \). So the equivalent statement in symbolic form is: \( p \wedge \sim q \). To really grasp these laws, think about their real-world applications, like simplifying logic circuits or even analyzing truth conditions in programming. When you swap between negation of conjunctions and disjunctions, you save time and avoid errors, making problem-solving way more efficient and neat! And if you're diving deeper into the world of logical statements, consider checking out resources like "Logic in Computer Science" by Huth and Ryan. It's a fun read filled with puzzles and examples that show logic's underpinnings in computer algorithms and systems. You’ll walk away with a strong understanding of how to use these concepts!

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