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In ordinary English (also natural language) "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. For example, being a man is a necessary condition for being a brother, but it is not sufficient—while being a man sibling is a necessary and sufficient condition for being a brother.
Wherever logic is applied, especially in mathematical discussions, it has the same meaning as above: it is an abbreviation for if and only if, indicating that one statement is both necessary and sufficient for the other. This is an example of mathematical jargon (although, as noted above, if is more often used than iff in statements of definition).
In other words, both the original and the general version of the pumping lemma give a necessary but not sufficient condition for a language to be regular. For example, consider the following language:
The symbol for the biconditional ("↔") signifies the relationship between the propositions is both necessary and sufficient, and is verbalized as "if and only if", or, according to the example "If P, then Q 'if and only if' if not Q, then not P". Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the ...
Necessary and sufficient condition, in logic, something that is a required condition for something else to be the case; Necessary proposition, in logic, a statement about facts that is either unassailably true (tautology) or obviously false (contradiction) Metaphysical necessity, in philosophy, a truth which is true in all possible worlds
Tests of sufficiency in biology are used to determine if the presence of an element permits the biological phenomenon to occur. In other words, if sufficient conditions are met, the targeted event is able to take place. However, this does not mean that the absence of a sufficient biological element inhibits the biological event from occurring.
For instance, enough money for a taxi implies that a minimum amount of money is necessary to pay for a taxi and that the amount of money in question is sufficient for the purpose. When functioning as determinatives in a noun phrase, sufficiency determiners select plural count nouns (e.g., sufficient reasons) or non-count nouns (e.g., enough money).
"It is not possible that X" is logically equivalent to "It is necessary that not X". Precisely what axioms and rules must be added to the propositional calculus to create a usable system of modal logic is a matter of philosophical opinion, often driven by the theorems one wishes to prove; or, in computer science, it is a matter of what sort of ...