Determine whether p,q and "If p, then q " are true or false in each case given below :
(P) 3 is a prime number

Answer

To determine the truth values of \( p \), \( q \), and the statement "If \( p \), then \( q \)" (denoted as \( p \implies q \)), we will follow these steps:

### Step 1: Define the statements \( p \) and \( q \)
- Let \( p \) be the statement "3 is a prime number."
- Let \( q \) be another statement that we need to define. For this example, let's say \( q \) is "2 is an even number."

### Step 2: Determine the truth value of \( p \)
- A prime number is defined as a number greater than 1 that has no positive divisors other than 1 and itself.
- The number 3 has the factors 1 and 3. Since it has exactly two distinct positive divisors, we conclude that \( p \) is **true**.

### Step 3: Determine the truth value of \( q \)
- The number 2 is indeed an even number, as it is divisible by 2 without a remainder.
- Therefore, \( q \) is also **true**.

### Step 4: Construct the truth table for \( p \implies q \)
- The implication \( p \implies q \) (if \( p \), then \( q \)) is true in the following cases:
1. If both \( p \) and \( q \) are true.
2. If \( p \) is false and \( q \) is true.
3. If \( p \) is false and \( q \) is false.

- The only case where \( p \implies q \) is false is when \( p \) is true and \( q \) is false.

### Step 5: Evaluate \( p \implies q \)
- Since both \( p \) (3 is a prime number) and \( q \) (2 is an even number) are true, we can conclude that \( p \implies q \) is **true**.

### Summary of Truth Values
- \( p \): True
- \( q \): True
- \( p \implies q \): True