## Necessary and Sufficient Conditions

There are other commonly used ways of expressing that proposition P implies proposition Q, or that P is equivalent to Q. Thus, if proposition P implies proposition Q, we state that P is a "sufficient condition" for Q. After all, for Q to be true, it is sufficient that P is true. Accordingly, we know that if P is satisfied, then it is certain that Q is also satisfied. In this case, we say that Q is a "necessary condition" for P. For Q must necessarily be true if P is true. Hence,

P is a sufficient condition for Q means: P => Q Q is a necessary condition for P means: P => Q

For example, if we formulate the implication in Example 1.7(c) in this way, it would read:

A necessary condition for x to be a square is that x be a rectangle.

A sufficient condition for x to be a rectangle is that x be a square.

The corresponding verbal expression for P Q is simply: P is a necessary and sufficient condition for Q, or P if and only if Q, or P iff Q. It is evident from this that it is very important to distinguish between the propositions "P is a necessary condition for Q" (meaning Q => P) and "P is a sufficient condition for Q" (meaning P => Q). To emphasize the point, consider two propositions:

1. Breathing is a necessary condition for a person to be healthy.

2. Breathing is a sufficient condition for a person to be healthy.

Evidently proposition 1 is true. But proposition 2 is false, because sick (living) people are still breathing. In the following pages, we shall repeatedly refer to necessary and sufficient conditions. Understanding them and the difference between them is a necessary condition for understanding much economic analysis. It is not a sufficient condition, alas!