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:

• 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.