# The Condition of Sufficiency and Necessity

Many a times, we are faced with the situation of finding the necessary condition and sufficient condition for some hypothesis to be true. Let’s consider a case (though it’s quite a specific case but you can generalize it) of scale space theory. We have to construct a scale-space representation of signal keeping in mind that fine scale information should be suppressed when the scale parameter increases. We also want that inherent property of the signal is retained at coarser scale. Along with it, we want no introduction of spurious structure in the process of smoothing. So, we have to formalize this concept. While formalizing this concept, we must know what conditions are absolutely necessary for construction of such space-scale representation of the signal. While deriving other conditions, we may find out that list goes on and on. So, the next question comes is where should it end? How should we describe sufficiency. We must know what is the condition for necessity and sufficiency. For this, we have to take the route for logic.

Let’s consider the statement, If “A number is divisible by $n(n+1)$” then “it is divisible by 2”. Let’s factorize this statement. S1: A number is divisible by $n(n+1)$ ; S2: it is divisible by 2. The clause S1 is called Hypothesis. It is what we are given, or what we may assume. The clause S2 is called Conclusion. Given the hypothesis, it is the statement that we must prove. When the If-then sentence is true, we say that the hypothesis is a sufficient condition for the conclusion. Thus it is sufficient to know that a number is divisible by $n(n+1)$ in order to conclude that it is divisible by 2. The conclusion is then called a necessary condition of that hypothesis; or necessary for that hypothesis to happen or be true. Because if a number is divisible by $n(n+1)$, it necessarily follows that it will be divisible by 2. (Equivalently, it is necessary for a number to be divisible by 2, if the number is to be divisible by $n(n+1)$).