https://file.aiquickdraw.com/mathbot/file/17275400949923d16di0s.jpg

Let's break down and solve the parts of this question using discrete math:

### Part (a) Truth Table for \( (p \rightarrow q) \lor (\neg p \rightarrow q) \):

First, recall the basic operations:
- \( p \rightarrow q \) is the implication: it is false only if \( p \) is true and \( q \) is false.
- \( \neg p \) is the negation of \( p \).
- \( \lor \) represents logical disjunction (OR), which is true if at least one of the propositions is true.

We'll construct the truth table step-by-step:

| \( p \) | \( q \) | \( p \rightarrow q \) | \( \neg p \) | \( \neg p \rightarrow q \) | \( (p \rightarrow q) \lor (\neg p \rightarrow q) \) |
|--------|--------|------------------------|--------------|---------------------------|-----------------------------------------------------|
|   T    |   T    |           T            |      F       |            T              |                         T                           |
|   T    |   F    |           F            |      F       |            F              |                         F                           |
|   F    |   T    |           T            |      T       |            T              |                         T                           |
|   F    |   F    |           T            |      T       |            F              |                         T                           |

- Column 3: \( p \rightarrow q \) is true except when \( p = T \) and \( q = F \).
- Column 4: \( \neg p \) is the negation of \( p \).
- Column 5: \( \neg p \rightarrow q \) is true except when \( \neg p = T \) and \( q = F \).
- Column 6: The disjunction \( (p \rightarrow q) \lor (\neg p \rightarrow q) \) is true whenever either of the previous implications is true.

### Part (b) Translation of the English Sentence into a Logical Expression:

The sentence is: “You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 15 years old.”

Let:
- \( p \): "You are under 4 feet tall."
- \( q \): "You are older than 15 years old."
- \( r \): "You can ride the roller coaster."

The sentence can be broken down into two parts:
1. "You cannot ride the roller coaster if you are under 4 feet tall."
   - This can be written as: \( p \rightarrow \neg r \) (If you are under 4 feet tall, you cannot ride).
   
2. "Unless you are older than 15 years old."
   - "Unless" translates to "if not." So, this can be written as \( \neg q \rightarrow \neg r \) (If you are not older than 15, you cannot ride).

Thus, the full logical expression becomes:
\[
(p \rightarrow \neg r) \land (\neg q \rightarrow \neg r)
\]
This means you cannot ride the roller coaster if either you are under 4 feet tall or not older than 15.

---

Let me know if you would like further details!

Here are five related questions you can explore:
1. What is the truth table for \( p \lor q \) (disjunction of \( p \) and \( q \))?
2. How does De Morgan’s Law apply to this proposition?
3. Can the logical expression in part (b) be simplified using any identities?
4. What is the equivalent of \( p \rightarrow q \) using only conjunctions and negations?
5. How would the truth table change if we replaced \( \lor \) with \( \land \) in part (a)?

**Tip**: Logical equivalences like De Morgan's Laws are key tools for simplifying complex propositions.

a) Construct the truth table for the compound proposition (p → q) ∨ (¬p → q). b) How can the English sentence 'You cannot ride the roller coaster if you are under 4 feet tall unless you are older than 15 years old.' be translated into a logical expression?

Learn how to construct a truth table for the logical expression (p → q) ∨ (¬p → q) and translate an English sentence into a logical expression. This problem covers fundamental concepts in propositional logic and logical equivalences, suitable for Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation