"Necessary and sufficient" is a phrase used in logic and mathematics to describe a relationship between two conditions or statements, indicating that one condition is required for the other to be true, and that when the first condition is present, it guarantees the truth of the second condition.
The term "necessary" refers to a condition that is indispensable or essential for the occurrence or truth of another condition. In this context, it means that the first condition must be present or satisfied in order for the second condition to exist.
On the other hand, "sufficient" signifies a condition that, once fulfilled or established, is adequate in itself to ensure the truth or occurrence of another condition. It implies that the first condition alone is enough to guarantee the truth of the second condition.
When combined, "necessary and sufficient" expresses a strong logical relationship between two conditions. It indicates that the first condition not only contributes to or influences the second condition but is also both required and capable of producing it. Thus, if the necessary condition is met, it is sufficient to guarantee the existence or truth of the related condition. If the necessary condition is not satisfied, the second condition cannot be true.
In summary, the phrase "necessary and sufficient" denotes an inseparable connection between two conditions, with the first being a prerequisite for the second, and the presence of the first being adequate to establish the truth or occurrence of the second.
