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  • If $p\implies q$ ("$p$ implies $q$"), then $p$ is a sufficient condition for $q$.

  • If $\lnot p\implies \lnot q$ ("not $p$ implies not $q$"), then $p$ is a necessary condition for $q$.

I don't understand what sufficient and necessary mean in this case. How do you know which one is necessary and which one is sufficient?

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    $\begingroup$ Those are the definitions of necessary and of sufficient. $\endgroup$
    – Zhen Lin
    Dec 11, 2012 at 9:37
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    $\begingroup$ Are you aware that $\bar p \implies \bar q$ is the logical equivalent of $q \implies p$ ? Do you know what "necessary" and "sufficient" mean in the English language? $\endgroup$
    – DanielV
    Nov 11, 2014 at 8:55

3 Answers 3

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Suppose first that $p$ implies $q$. Then knowing that $p$ is true is sufficient (i.e., enough evidence) for you to conclude that $q$ is true. It’s possible that $q$ could be true even if $p$ weren’t, but having $p$ true ensures that $q$ is also true.

Now suppose that $\text{not-}p$ implies $\text{not-}q$. If you know that $p$ is false, i.e., that $\text{not-}p$ is true, then you know that $\text{not-}q$ is true, i.e., that $q$ is false. Thus, in order for $q$ to be true, $p$ must be true: without that, you automatically get that $q$ is false. In other words, in order for $q$ to be true, it’s necessary that $p$ be true; you can’t have $q$ true while $p$ is false.

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    $\begingroup$ As usual, I will point out that a downvote not accompanied by an explanatory comment is less than useful. $\endgroup$ Dec 11, 2012 at 12:46
  • $\begingroup$ how do you know that p is true or not? $\endgroup$ Dec 12, 2012 at 8:44
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    $\begingroup$ @cloud9resident: You don’t know. Whether $p$ is true or not is a completely separate issue from whether it is a necessary or sufficient condition for $q$. The latter does not depend in any way on the truth or falsity of $p$. $\endgroup$ Dec 12, 2012 at 8:46
  • $\begingroup$ But for not p implies not q, in terms of that reasoning, not q doesn't tell you anything about not p because here are different paths to not q? $\endgroup$ Jan 25, 2015 at 22:32
  • $\begingroup$ @committedandroider: That’s correct. Knowing that $\neg q$ is true tells you nothing about whether $\neg p$ is true or not. $\endgroup$ Jan 25, 2015 at 22:34
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I always think of it in terms of sets.

enter image description here

In the picture above, for an element to be purple, it's necessary to be red, but it is not sufficient.

The same holds for the blue set, to be in the blue set is a necessary condition in order to be purple, but it is not enough, it's not sufficient.

A sufficient condition is stronger than a necessary condition. If you tell me that you have a red or blue element I can't say for sure if it is in the purple set, but if you tell me that you have a purple element I now for sure that it is in the red and blue sets.

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  • $\begingroup$ But purple is neither red nor blue… $\endgroup$
    – Geremia
    Aug 30, 2018 at 15:31
  • $\begingroup$ Yes, and this comment is an edit to change "have a purple element" to "have an element from the purple area. I am so happy to read something that does not use the word imply. $\endgroup$
    – user756686
    Mar 23, 2020 at 14:35
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It has always been difficult for me to connect the linguistic sense of the terms "necessary" and "sufficient" with mathematics. However, with practice with the following table, this skill becomes automatic.

enter image description here

I'm assuming you understand the meaning of the implication. In any case it is helpfull to think about simple example when looking at this table. For example: "If it is raining, then close the window". Then it remains to understand the following.

P is a sufficient for Q. If P is true then Q will be always true (the first line in the table). Note that we do not consider the second line. But as we see in the table Q can be true also when P is false (the third line in the table). So P is "just" a sufficient condition for Q.

Q is a necessary condition for P. It is obvious from the table. P can be true "only" when Q is true (the first line in the table). Note that we do not consider the second line.

About $\lnot A \implies \lnot B$. It is called the contrapositive of the statement $A\implies B$. As you see it is not necessary for you for understanding the "sufficiency" and "necessaty".

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