Question

“Where there is smoke, there is fire.” Which of the following statements, if true, would show that the above statement is false?

• A. There is sometimes smoke where there is no fire.
• B. There is sometimes fire where there is no smoke.
• C. There is no fire where there is no smoke.
• D. None of the above.

Hint:

To disprove a universal statement like “All A implies B,” find one case where A happens but B does not. This is called a counter-example.

Answer 1

The Correct Option is A

The original statement is: \[ \text{“Where there is smoke, there is fire.”} \] This implies that smoke \(\Rightarrow\) fire. It is a universal conditional statement. To prove this statement false, we need a counter-example. That is, a situation where: \[ \text{Smoke is present, but fire is not.} \] Let's evaluate the options: (a) “There is sometimes smoke where there is no fire.” This directly contradicts the statement. It provides a counter-example where smoke exists, but fire does not. Hence, this makes the statement false. (b) “There is sometimes fire where there is no smoke.” This does not contradict the statement, because the statement doesn’t say fire only occurs with smoke. This is irrelevant. (c) “There is no fire where there is no smoke.” This aligns with the original statement, not contradicts it. So this supports, not falsifies, the statement. (d) “None of the above.” Incorrect because (a) does provide a valid contradiction. % Final Answer \[ \boxed{\text{(a)}} \]