We were having a discussion at the office when we should have been working and I suggested that an example of a Sufficient but not Necessary condition is:
Given a natural number of fewer than, say, 25 digits.
We wish to establish if it is divisible by six.
An example of a Necessary but not Sufficient condition was: Is the number divisible by two? The other guy was happy with that and agreed.
My example of a Sufficient but not Necessary condition (however simple) was:
Is the number equal to six? The other guy (who has a degree in math) insisted that because this would not apply to numbers such as 12 this could not be Sufficient. I maintained that that is the whole point, the condition is not necessary but because it is Sufficient, if it is true, you are done, QED.
This promptly devolved into "It is not", "It is, too" which did not seem very mathematical, somehow.
Could we get somebody to comment? The other guy decided he did not want to discuss this further but I would like to feel a little vindicated. (If I am wrong, I will send him your answer.) Thank you in advance.
Best Answer
$A$ is necessary for $B \iff B \implies A$
$A$ is sufficient for $B \iff A\implies B$
In view of this, let's check.
Case 1: $A:$number is divisible by $2$, $B:$ number is divisible by $6$.
Does $B \implies A?$... definitely!! So $A$ is necessary for $B$.
Does $A\implies B?$.... well not always. So $A$ is not sufficient for $B.$
Case 2: $A:$number is $6$, $B:$ number is divisible by $6$.
Does $B \implies A?$... well not always!! So $A$ is not necessary for $B$.
Does $A\implies B?$.... definitely. So $A$ is sufficient for $B.$
You were right!!