Translate "A is not necessary for B" in math expression

Question

I tried to to solve this but what I found is

A is not necessary for B

I could be wrong

= not(A is necessary for B)
= not(not(B) or A)
= not(A) and B

but it doesn't make sense. Let's take an example:

A = understand things
B = argue about things
A is not necessary to B = not(A) and B

so

Understand things is not necessary to argue about things = Not understand things and argue about things

should be the same thing. I really appreciate any kind of help.

Answer 1

"$A$ necessary for $B$" is actually "$B$ implies $A$"

So "$A$ not necessary for $B$" is actually "$B$ does not imply $A$"

In mathematical notation:$$\neg[B\implies A]\text{ or equivalently }\neg A\wedge B$$

This states that $B$ can be true while at the same time $A$ is not true (so no necessity for $A$ to be true in order to achieve that $A$ is true).

Answer 2

When you interpret "is necessary" as "is implied by" then indeed we have "$A$ is not neccessary for $B$" exactly where "not $A$ yet $B$".

Check the truth table: $$\begin{array}{l:l|cc}A & B & A\gets B\\\hline \top & \top & \top \\ \top & \bot & \top \\ \bot & \top & \bot &\star \\\bot& \bot & \top \end{array}$$

---

The issue is you are interpreting "is not necessary" as "is maybe not implied by"; which requires modal logic quantifiers.

$$\neg\Box(B\to A)\equiv \Diamond(B\land\lnot A)$$