Necessary and/or SufficientDate: 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/ |
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