Given below are two statements :
p : 30 is a multiple of 5.
q : 30 is a multiple of 9.
Write the compound statement, connecting these two statements with ' and ' and 'or ' . In both cases, check the validity of the compound statement.

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the Statements
We have two statements:
- Statement \( p \): "30 is a multiple of 5."
- Statement \( q \): "30 is a multiple of 9."

### Step 2: Write the Compound Statement Using 'AND'
The compound statement using 'AND' can be written as:
- "30 is a multiple of 5 AND 30 is a multiple of 9."

### Step 3: Check the Validity of the 'AND' Statement
To check the validity of the compound statement using 'AND':
- We know that 30 is a multiple of 5 (since \( 30 \div 5 = 6 \)).
- However, 30 is not a multiple of 9 (since \( 30 \div 9 \) does not yield an integer).

Thus, the compound statement "30 is a multiple of 5 AND 30 is a multiple of 9" is **FALSE** because one part of the statement (30 is a multiple of 9) is false.

### Step 4: Write the Compound Statement Using 'OR'
The compound statement using 'OR' can be written as:
- "30 is a multiple of 5 OR 30 is a multiple of 9."

### Step 5: Check the Validity of the 'OR' Statement
To check the validity of the compound statement using 'OR':
- As established, 30 is a multiple of 5 (TRUE).
- 30 is not a multiple of 9 (FALSE).

In an 'OR' statement, if at least one part is true, the whole statement is considered true. Since 30 is a multiple of 5, the compound statement "30 is a multiple of 5 OR 30 is a multiple of 9" is **TRUE**.

### Summary of Results
- The compound statement using 'AND' is **FALSE**.
- The compound statement using 'OR' is **TRUE**.