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### Necessary and/or Sufficient

```Date: 05/26/2002 at 09:13:35
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

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/
```
Associated Topics:
College Logic
High School Logic

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