# A+B is sufficient and necessary for C. C is unnecessary for D. Is it flawless to formally write that "A and B are unnecessary condition for D"?

Condition A+B is sufficient and necessary for condition C. Condition C is sufficient but not necessary for D.

Is it flawless to write that: "A and B are unnecessary condition for D "?

I am afraid that this could be a non-rigorous sentence as condition A could be necessary for condition D. The bolded claim above could also be sloppy as the grammar seems mistaken. If instead I wrote "A and B are unnecessary conditions for D", people may plausibly interpret it as "each of A and B are not necessary for D separately".

Is "unnecessary condition" a legitimate mathematical term? If not, what term shall I use?

The question is more about terminology, definitions, grammar, and the use of phrases.

Update: my "$$+$$" meant conjunction. However, thank to Aman Kushwaha, I just note that the plus sign "+" is rarely used by mathematicians as the conjunction operation. In fact, after some researches, I found that "+" occasionally means OR, the symbol "$$\lor$$".

• Does "condition A+B" mean "condition A and condition B"? Oct 16 at 17:02
• @AmanKushwaha I am not sure about the English grammar but I think "condition A+B" means that both A and B must be satisfied. Oct 16 at 17:03
• @RyanG Thank you very much Ryan! Your explanation indeed makes sense. Oct 16 at 22:58
• Your premises are (1) $(A\&B)\leftarrow\rightarrow C$ and (2) $\neg(\neg C\rightarrow\neg D)$. Premise (2) can be rephrased as $\neg (D\rightarrow C)$ , which in turn is equivalent to $D \& \neg C$ , implying that $C$ is false . So from (1) and (2), by modus tollens , one can conclude that either A is false or B is false. Oct 17 at 10:15

I agree that "are unnecessary condition" is incorrect grammar. In the appropriate contexts one could write "is an unnecessary condition" or "are unnecessary conditions". In this particular context, if you write "conditions" instead of "condition" I agree that people will consider $$A$$ and $$B$$ separately as possible conditions for $$D$$. If you write "is", people might realize you mean the conjunction "$$A$$ and $$B$$" as a single condition or they might think you have a mistake in your grammar and misunderstand you.

The difficulty as I see it is that if you start a sentence with the phrase "$$A$$ and $$B$$," it is difficult to communicate the fact that you mean a single condition as opposed to a list of two separate conditions. Some ways to avoid this problem might be to write the condition in one of these ways:

• "The condition '$$A$$ and $$B$$'."
• "The conjunction of $$A$$ and $$B$$."
• $$A \land B$$.

I think the term "unnecessary condition" sometimes occurs in the sense you intend it here, but if you want to avoid it, you could write "is not a necessary condition" instead.

Aside from these concerns about how you make the statement you are trying to make, the statement itself is fine.

• Many thanks David I agree with you. May ask if $A+B$ is the same as $A\land B$? Oct 16 at 17:08
• @HighGPA I know the question is not asked to me but according to your previous reply to my comment, I'd say yes, your way of seeing $A+B$ is same as $A \land B$. Oct 16 at 17:11
• @AmanKushwaha Thanks Aman for your answer. Of course you are more than welcome to answer any of my questions. I wonder if the way of how I was using "$A+B$" is rigorous enough? Or, people might think my notation "$+$" confusing and sloppy. Oct 16 at 17:13
• It depends on what has been written previously in the same document. If there was a definition of notation for logical conjunction using the symbol $+$ (either explicitly or implicitly) then you can certainly use it. If not, then "$A+B$" in this context might still be understood correctly (and would implicitly define $+$ as conjunction). To make sure it is understood, you could write, "The conjunction $A+B$." I prefer $\land$ or $\&$; I don't think there's an ironclad rule that you can't use $+$ for conjunction, but I think I've seen it used for disjunction, so you need to make it clear. Oct 16 at 17:14
• @DavidK Thank you again David! I think the terms you used are perfect without confusion or sloppiness. I will use those terms and phrases. Oct 16 at 17:16

Is it flawless to write that: "A and B are unnecessary condition for D "?

This description is ambiguous: "condition" being singular suggests that A and B are being referred to collectively, yet "are" contradicts this.

The given information is this: $$(A+B)\implies D.$$ If $$+$$ means conjunction, then I would write "A and B collectively being true is an unnecessary condition for D to be true", which I think is fine to shorten as "A and B is an unnecessary condition for D". ("Unnecessary" is fine, since necessity in this context has only two states: necessary versus not necessary.)

Remarks in response to your Update:

Union $$\cup$$ of sets is analogous to and disjunction $$\lor$$ of propositions, just as intersection $$\cap$$ and conjunction $$\land$$ are analogous. For mutually exclusive events, $$P(A\cup B) =P(A)+P(B);$$ for independent events, $$P(A\cap B)=P(A)\times P(B).$$