Give an example of a condition that is sufficient but not necessary?

Consider the Mobius mu function. A sufficient but not necessary condition for mu(n) = -1 is for n to be prime. If n is composite but has three distinct prime factors, then mu(n) = -1 as well.

I imagine you know lots of examples of conditions that are necessary but not sufficient, e.g., for 2^n - 1 to be prime, it is necessary that n also be prime, but that does not guarantee that 2^n - 1 is actually prime.

Necessary And Sufficient Conditions Examples

Sufficient Conditions:

A key idea is that SUFFICIENT conditions GUARANTEE that a particular consequence must follow.

Sufficient but not Necessary: This condition guarantees the consequence but alternate conditions might also get the required result.

Example 1: Suppose that 8 or more out of 10 correct logic questions is needed to get grade A in a test.

We can say that gaining 9 out of 10 logic questions correct is a sufficient condition for earning grade A. But we cannot say that if you do not have 9 out of 10 you can't have an A, because it is not necessary to get 9. Since 8 out of 10 would also get an A, we say that gaining 9 out of 10 is a sufficient but not necessary condition.

Example 2: Heavy rainfall is a sufficient condition for the ground to be wet. But it is not necessary for it to be raining for the ground to be wet. The garden sprinkler could be on.

In neither of these two examples is the sufficient condition also a necessary condition.

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Contrast that with:-

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Necessary but not Sufficient: The one condition being spoken of is needed, but additional conditions may also be needed.

Example 1: Having oxygen in the earth's atmosphere is a necessary condition to sustain human life.

But note that just having oxygen will not guarantee human life. There are many other conditions, (food, water, shelter etc.) needed to sustain human life other than oxygen in the atmosphere.

Example 2: Having fuel in a car is a necessary condition to get it started. But there are many other requirements, (having a battery, starter motor etc.).

Regards - Ian

That's unclear and there are basically several answers to the question...

If it is not necessary, then it is not condition at all.

Think yourself.. anyhow good try :-)