Necessary and/or Sufficient

Date: 05/26/2002 at 09:13:35
From: Adeel
Subject: logic

What do we mean by 'necessary condition' and 'sufficient condition' 
(and sometimes we call a condition both 'necessary and sufficient')? I 
am very much confused. Help!

Date: 05/28/2002 at 09:15:25
From: Doctor Peterson
Subject: Re: logic

Hi, Adeel.

Let's look at the two statements (predicates), "X is a mammal" and "X 
is a dog". Call the first statement A, and the second B.

Now, A is a _necessary_ condition for B, because A _must_ be true in 
order for B to be true. B can only be true if A is true; if A is not 
true, then B can't be true. We can say this in several ways:

    A is a necessary condition for B

    A <== B      (A is implied by B)

    B ==> A      (B implies A)

    A if B       (whenever B is true, A will be true)

    B only if A  (B is true only when A is true)

On the other hand, A is not a _sufficient_ condition for B, which 
would mean that in order to know that B is true, it is _enough_ to 
know that A is true. It is not enough to know that X is a mammal, 
because there are other mammals besides dogs. But if we reverse the 
two statements, we find that B is a sufficient condition for A: if we 
know that X is a dog, we know that it is a mammal. So these 
statements are equivalent:

    A is a sufficient condition for B

    B <== A      (B is implied by A)

    A ==> B      (A implies B)

    B if A       (whenever A is true, B will be true)

    A only if B  (A is true only when B is true)

Note that "necessary condition" and "sufficient condition" are 
opposites; "A is a necessary condition for B" means the same thing 
as "B is a sufficient condition for A".

Now, if A is a necessary AND sufficient condition for B, then the 
implication works both ways; it can be expressed as

    A <==> B  (A is equivalent to B)

    A iff B   (A if and only if B)

This means that if A is true, B must be true, and if B is true, A 
must be true. That is not the case in our example statements; but it 
would be true, for example, if A were "X is less than Y" and B were 
"Y is greater than X". These two statements mean the same thing; 
if one is true, then the other is true. So if we want to prove B, it 
is necessary for A to be true, and it is sufficient to prove that A 
is true.

Here is a discussion of these concepts from our archives:

    http://mathforum.org/library/drmath/view/55668.html

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/