Secolinsky.com latest post feedhttp://www.secolinsky.com/rss/The latest posten-usThu, 24 Jun 2021 04:23:12 -0000Necessary and Sufficient in Mathematicshttp://example.com/blogs/necessary-and-sufficient-conditions,25/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.
---snippet---
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.
---endsnippet---
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 \(B\):
\[
\left( A \Leftrightarrow B \right ) \Leftrightarrow \left( A \text{ is necessary and sufficient for } B \right).
\]
http://example.com/blogs/necessary-and-sufficient-conditions,25/