Sufficient but not Necessary conditions

Question

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.

Answer 1

$A$ 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!!

Answer 2

You are right. The condition A is said to be sufficient for an event B if (and only if) the occurence of A guarantees the occurence of B. The logical relation would be

$A\Rightarrow B$

Then A is by no means necessary for B: $B\Rightarrow A$ or even a characterization of B: $A \Leftrightarrow B$

So for example, to be a sheep (6) is sufficient for being an animal (divisible by 6). However you don't have to be a sheep (6) in order to be an animal (divisible by 6).

Answer 3

Per this answer, 'A is sufficient for B.`   means that    'A is a subset of B'.

A picture and real-life example may aid to understand the following:

Your main question is whether (2) follows from (1);

(1) P → Q [ P is a sufficient condition for Q ],

(2) Q → P [ P is a necessary condition for Q ].

The reason the entailment from (1) to (2) doesn't hold is that it's possible that Q follow from some proposition R that is not equivalent to P. The only instance where the entailment is realized is one where all necessary conditions for Q are logically equivalent to P.

The above is exemplified in the picture below, if P = Northern Ireland, Q = UK, R = Great Britain.

Then being in Northern Ireland is sufficient for being in the UK,
but is NOT necessary for being in the UK, because one can also be in U.K. by being in
Great Britain.

Answer 4

"$n=6$" is a sufficient condition for a number $n$ to be divisible by $6$. That's because $$ n=6\implies n\text{ is divisible by }6 $$ is true.