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"?

Question

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?

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

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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 "$\lor$".

Answer 1

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.

Answer 2

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.)

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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, $P(A\cap B)=P(A)\times P(B).$