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Let p be the statement "x is divisible b...

Let p be the statement "x is divisible b, 4" and q be the statement "x is divisible by 2".
STATEMENT-1 : `p hArr q`
and
STATEMENT-2 : If x is divisible by 4, it must be divisible by 2 .

A

Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1

B

Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1

C

Statement-1 is True, Statement-2 is False

D

Statement-1 is False, Statement-2 is True

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the two statements given, which involve the divisibility of a number \( x \) by 4 and 2. ### Step-by-Step Solution: 1. **Define the Statements:** - Let \( p \) be the statement "x is divisible by 4". - Let \( q \) be the statement "x is divisible by 2". 2. **Analyze Statement 1: \( p \iff q \) (p if and only if q):** - This means that \( p \) is true if and only if \( q \) is true. - In logical terms, this can be broken down into two parts: - If \( p \) is true, then \( q \) must be true (if \( x \) is divisible by 4, then \( x \) is divisible by 2). - If \( q \) is true, then \( p \) must be true (if \( x \) is divisible by 2, then \( x \) is divisible by 4). 3. **Check the First Part:** - If \( x \) is divisible by 4, it can be expressed as \( x = 4k \) for some integer \( k \). - Since \( 4k \) is clearly divisible by 2 (as \( 4k = 2(2k) \)), this part is true. - Therefore, \( p \) implies \( q \) is true. 4. **Check the Second Part:** - If \( x \) is divisible by 2, it can be expressed as \( x = 2m \) for some integer \( m \). - However, \( x \) being divisible by 2 does not guarantee that it is divisible by 4. For example, if \( x = 6 \) (which is divisible by 2), it is not divisible by 4. - Therefore, \( q \) does not imply \( p \) is false. 5. **Conclusion for Statement 1:** - Since the second part of the biconditional \( p \iff q \) is false, we conclude that Statement 1 is false. 6. **Analyze Statement 2:** - Statement 2 says, "If \( x \) is divisible by 4, it must be divisible by 2". - As established earlier, if \( x = 4k \), then \( x \) is also divisible by 2. - Therefore, this statement is true. 7. **Final Conclusion:** - Statement 1 is false, and Statement 2 is true. ### Final Answer: - Statement 1 is false, and Statement 2 is true.
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