Necessary vs. Sufficient condition

I always get confused between necessary and sufficient conditions. So here’s a small picture to remind myself:

Suppose we care about condition A. It is very important to us. But there is condition N which is a superset of condition A. And there is condition S which is a subset of condition A.

N is necessary for A. No element in A, is not also a part of N.

S is sufficient for A. All elements in S must be also in A.

N is NOT sufficient for A. There are elements in N which are not in A (e.g. n).

S is NOT necessary for A. There are elements in A which are not in S (e.g. a).

If I want to prove that something is IN A, it might be easier to prove that it’s in S, and as S is sufficient, I’m done. If I want to prove that something is NOT IN A, it might be easier to prove that it’s not in N, and so it doesn’t fulfill the necessary condition.

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