Consider the following statement: You loose weight if and only if you do sports. Which of the following is correct for this statement?
A) Doing sports is not a necessary condition for loosing weight.
B) Doing sports is not a sufficient condition for loosing weight.
C) None of the above
D) Doing sports is both a necessary and sufficient condition for loosing weight.
The correct answer and explanation is:
The correct answer is D) Doing sports is both a necessary and sufficient condition for losing weight.
Explanation:
In logic, the phrase “if and only if” (often abbreviated as “iff”) represents a bidirectional condition. This means that both conditions must be true at the same time, or both false at the same time, for the overall statement to hold.
The statement “You lose weight if and only if you do sports” can be broken down into two parts:
- You lose weight if you do sports. This implies that doing sports is a sufficient condition for losing weight.
- You lose weight only if you do sports. This implies that doing sports is a necessary condition for losing weight.
Necessary Condition:
A necessary condition is something that must be true for a particular outcome to happen. In this case, the statement says that if you lose weight, it must be because you are doing sports. Therefore, doing sports is necessary for losing weight.
Sufficient Condition:
A sufficient condition is something that, if true, guarantees a particular outcome. The statement also implies that if you are doing sports, then you will lose weight. Therefore, doing sports is sufficient for losing weight.
Since both conditions—doing sports being necessary and sufficient for losing weight—are true according to the given statement, the correct answer is that doing sports is both a necessary and sufficient condition for losing weight.
This kind of logical structure is central to understanding how two conditions can be intertwined, as both must happen together or neither will occur.