Download Comments on the Meaning of Necessary and Sufficient Conditions

Some Thoughts on Logic
N. Guydosh
10/19/03
In mathematics our understanding of the terms necessary conditions and sufficient
conditions may have significant on how to properly use theorems in solving problems.
The proper use of proof by contradiction depends on out understanding of these terms. A
good example is the use of definitions, the pumping lemma, or closure theorems to
determine if a language is regular of not. Math books sometimes use these terms without
formally defining them. This document will define and give some interpretations to these
terms.
Before starting a good reference on these concepts is “Discrete Mathematics and Its
Applications”, 4th ed., pp. 5-6, by Kenneth H. Rosen.
Let the symbol “==>” mean “imply”, for example P ==> Q means P implies Q.
Intuitive meaning
Let A and B be truth statements or “events” that may possible affect each other.
“A is a sufficient condition for B” means that if statement A is true (or event A is
present), then statement B is true (or event B is present). In symbolic logic this is
A ==> B. The conclusion is guaranteed to be true if the premise (sufficient condition) is
true.
“A is a necessary condition for B” means that if statement A is false (or event A is
absent), then statement B is false (or event B is absent). In symbolic logic this is
B ==> A. The premise is guaranteed to be false if the conclusion (necessary condition)
is false.
The above observation is a simple consequence of the equivalence the two statement:
A==> B and the statement B’ ==> A’. Where the “prime” means complement.
More formal meaning
In symbolic logic, Let P and Q be truth statements with a value of either true or false.
If P ==> Q
then
P is a sufficient condition for Q
and
Q is a necessary condition for P
if P ==> Q and Q ==> P, ie., P <==> Q, (P if and only if Q) then P is necessary and
sufficient for Q and vice versa.
Set theory interpretation
If P and Q are sets, then If P ==> Q becomes P  Q.
Which means (by definition) that x  P ==> x  Q. Thus being in a subset is a is a
necessary condition for being in the superset. But also x  Q implies that x  P, and
being in Q is a necessary condition for being in P. Necessary and sufficient in set theory
would mean that P and Q are identical.
Some Applications
Definitions are generally if and only if statements. Theorem 3.5 states that language L is
regular if and only if there exists a regular grammar (left linear) defining the language.
The existence of a regular grammar is necessary and sufficient for the L to be regular.
But note that it does not say that all grammars that define L must be regular. Careful of
the wording of the theorem! Thus if grammar G is regular, then L is regular, but if L is
regular we cannot conclude that all defining G’s are regular. We can only conclude that
there exists a defining G that is regular. This all boils down to mean that all regular
grammars define regular languages, but there may exist some non-regular grammars that
also define regular languages. For example:
The regular language L = L(a*b*) has a required regular grammar:
S-> aS | B | , B -> bB | b
but it also is defined by a non-regular grammar:
S -> aSb | aS | Sb | .
The Regular Pumping Lemma essentially says if L is regular, then it can be “pumped”
(according to Theorem 4.8). Then being regular is a sufficient condition for “staying
regular under pumping”, and staying regular under pumping is a necessary condition for
being regular. One way of looking at the pumping lemma, is that we show that a
necessary condition is false:
L is regular ==> Pumping lemma works
But we show that the pumping lemma fails (necessary condition is false)
Thus the premise must be false ie., L is not regular … only false can imply false