premise - A reason that supports the conclusion in an argument

Necessary and Sufficient in Mathematics

By Maurice Ticas

Logic

A quick search on the internet for the term "predicate logic" gives a definition from wiki as its first result: "[A] predicate is the formalization of the mathematical concept of statement. A statement is commonly understood as an assertion that may be true or false, depending on the values of the variables that occur in it." Rigorously studying mathematics requires a good command of predicate logic. This is because proofs are comprised of statements that lead us to develop theory. Now in our journey of whatever mathematical theory we participate, we can sometimes express a relationship among two statements A and B as saying one statement is necessary and/or sufficient for the other statement.

We'll now break it down what it means to say that such a relationship is necessary and/or sufficient.

Let's have two statements $$A,B$$ be given. To say that $$A$$ implies $$B$$ is equivalent to saying that $$A$$ is sufficient for $$B$$; likewise, it is also equivalent to saying that $$B$$ is necessary for $$A$$. If we are to express in symbols, we'd write that $\left( A \Rightarrow B \right ) \Leftrightarrow \left(A \text{ is sufficient for } B \right) \Leftrightarrow \left(B \text{ is necessary for } A\right).$

In symbols, it is easy to write what it means to say that $$A$$ is necessary and sufficient for $$B$$: $\left( A \Leftrightarrow B \right ) \Leftrightarrow \left( A \text{ is necessary and sufficient for } B \right).$