Determine whether p,q and "If p, then q " are true or false in each case given below :
`3(5 -: 6 ) lt 1`

Answer

To determine whether the statements \( p \), \( q \), and "If \( p \), then \( q \)" are true or false, we will first define \( p \) and \( q \) based on the given inequality \( 3(5 -: 6) < 1 \).

### Step 1: Define \( p \) and \( q \)
Let:
- \( p \): \( 3(5 -: 6) < 1 \)
- \( q \): A statement we need to evaluate based on \( p \).

### Step 2: Evaluate \( p \)
We need to evaluate \( p \):
\[ 3(5 -: 6) < 1 \]
This can be rewritten as:
\[ 3 \left(\frac{5}{6}\right) < 1 \]
Calculating \( 3 \times \frac{5}{6} \):
\[ \frac{15}{6} = 2.5 \]
Now, we check if \( 2.5 < 1 \):
This is **false** since \( 2.5 \) is greater than \( 1 \).

Thus, we conclude:
- \( p \) is **false**.

### Step 3: Define \( q \)
Since the problem does not specify \( q \), we will assume \( q \) is a statement that can be true or false based on the context. For the sake of this problem, let's assume \( q \) is a statement that is also **false**.

### Step 4: Evaluate "If \( p \), then \( q \)"
The implication "If \( p \), then \( q \)" (denoted as \( p \implies q \)) is evaluated based on the truth values of \( p \) and \( q \):
- If \( p \) is **false** and \( q \) is **false**, then \( p \implies q \) is **true**.

### Summary of Truth Values
- \( p \): **false**
- \( q \): **false**
- \( p \implies q \): **true**

### Final Result
- \( p \) is false.
- \( q \) is false.
- \( p \implies q \) is true.