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To solve this problem using **De Morgan's laws**, we first need to understand the given sentence:

- The sentence is: "It is false that the band plays rock music and the band has a guitarist."

### Step 1: Express the sentence in symbolic logic
Let:
- \( P \) = "The band plays rock music."
- \( Q \) = "The band has a guitarist."

The sentence "The band plays rock music and the band has a guitarist" can be written as:
- \( P \land Q \).

Now, the sentence states that **it is false** that \( P \land Q \). This can be written as:
- \( \neg (P \land Q) \).

### Step 2: Apply **De Morgan's law** to \( \neg (P \land Q) \)
According to De Morgan's law, \( \neg (P \land Q) \) is equivalent to:
- \( \neg P \lor \neg Q \).

In words, this becomes:
- "The band does not play rock music **or** the band does not have a guitarist."

### Step 3: Match this with the given options
Now, let's review the answer choices based on this logical expression:

- **First option**: "If the band does not play rock music, then the band does not have a guitarist."  
  This is an incorrect conditional statement.

- **Second option**: "The band plays rock music or the band does not have a guitarist."  
  This contradicts the original statement, so it's incorrect.

- **Third option**: "The band does not play rock music and the band does not have a guitarist."  
  This is not the correct interpretation either, as it involves an "and" instead of an "or."

- **Fourth option**: "The band does not play rock music or the band does not have a guitarist."  
  This matches the result of applying De Morgan's law and is the correct answer.

### Correct Answer: **Fourth option**

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Would you like more details on De Morgan's laws or the reasoning behind any of the incorrect options?

### Follow-up Questions:
1. Can De Morgan's laws be applied to other logical connectives like "or"?
2. What are some examples of De Morgan's laws in everyday language?
3. How would the answer change if the statement was about the band not playing rock music?
4. Can you apply De Morgan's law to a conditional statement?
5. How do De Morgan's laws relate to truth tables?

**Tip:** De Morgan's laws help simplify complex logical expressions, especially when dealing with negations in compound statements.

Use De Morgan's laws to write an equivalent statement for the given sentence: It is false that the band plays rock music and the band has a guitarist.

Learn how to apply De Morgan's Laws to transform a negated conjunction into a disjunction. The problem involves using symbolic logic to rewrite the sentence: 'It is false that the band plays rock music and the band has a guitarist.' Suitable for students in Grades 10-12.

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