Necessary and sufficient conditions

Conditions for the validity of a proposition $A$ without which $A$ cannot possibly be true (necessary conditions), while when these conditions are satisfied, then $A$ must be true (sufficient conditions). Frequently the expression "necessary and sufficient" is replaced by "if and only if". Necessary and sufficient conditions have great significance. In complex mathematical problems the search for necessary and sufficient conditions that are convenient to use, sometimes becomes extremely difficult. In such cases one tries to find sufficient conditions that may be wider, that is, comprise possibly more cases in which the fact of interest still holds, and necessary conditions that may be narrower, that is, comprise possibly fewer cases in which the relevant fact does not hold. In this way, sufficient conditions come close step-by-step to necessary ones.