Fill in each blank so that the resulting statement is true.

De​ Morgan's laws state that​ ~ (p 

logical and∧ 

​q) 

equivalent≡ 

​_____ and​ ~ (p 

logical or∨ 

​q) 

equivalent≡ 

​_____.

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Part 1

De​ Morgan's laws state that​ ~ (p 

logical and∧ 

​q) 

equivalent≡ 

▼
 
tilde p logical or tilde q~p ∨ ~q

tilde p logical and tilde q~p ∧ ~q
 
and​ ~ (p 

logical or∨ 

​q) 

equivalent≡ 

▼
 
tilde p logical and tilde q.~p ∧ ~q.

tilde p logical or tilde q.

De Morgan's laws are important principles in logic, which relate conjunctions (AND) and disjunctions (OR) with negations. According to De Morgan's laws:

1. \(\sim (p \land q) \equiv (\sim p \lor \sim q)\)
   - The negation of \(p \land q\) (logical AND) is equivalent to \(\sim p \lor \sim q\) (NOT \(p\) or NOT \(q\)).

2. \(\sim (p \lor q) \equiv (\sim p \land \sim q)\)
   - The negation of \(p \lor q\) (logical OR) is equivalent to \(\sim p \land \sim q\) (NOT \(p\) and NOT \(q\)).

So, the complete statements would be:

1. De Morgan's laws state that \(\sim (p \land q) \equiv (\sim p \lor \sim q)\).
2. De Morgan's laws state that \(\sim (p \lor q) \equiv (\sim p \land \sim q)\).

### Explanation:

- The first law says the negation of an AND statement becomes an OR statement with negated terms.
- The second law says the negation of an OR statement becomes an AND statement with negated terms.

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Do you want any further details or have any questions about De Morgan's laws?

Here are five related questions for practice:

1. What is the negation of \(p \land \sim q\) using De Morgan’s laws?
2. How would you apply De Morgan's laws to \(\sim (p \lor \sim q)\)?
3. What is the negation of \(\sim (p \land \sim q)\)?
4. How are De Morgan’s laws applied in set theory?
5. Can De Morgan’s laws be extended to more than two variables?

**Tip**: Remember, De Morgan's laws are useful for simplifying complex logical expressions by distributing negations!

Fill in each blank so that the resulting statement is true. De Morgan's laws state that ~ (p ∧ q) ≡ _____ and ~ (p ∨ q) ≡ _____.

Explore De Morgan's laws with a detailed explanation and a fill-in-the-blanks exercise. This problem covers logical concepts such as negation, conjunction, and disjunction, demonstrating the transformation of logical expressions using De Morgan's laws. Suitable for high school students in Grades 9-12.

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