I think is "Any of the sentence is wrong" but I'm not sure, maybe "Any of the sentences is right"?
2026-04-18 08:23:15.1776500595
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What's the negation of "One of the sentence is false"?
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If we interpret the word "one" as meaning "exactly one" then negating "one of the sentences is false" means that we should not have exactly one of the sentences false thus we should have the sentence
$$\text{All sentences are true or more than one sentence is false}$$
We could of course interpret "one" as meaning "at least one" (something which may be argued to be the most common mathematical interpretation) in which case we would translate the sentence to $$\text{All sentences are true }$$ Which is what you answered (formulated a bit differently).
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Let $F(x)$ mean that the sentence $x$ is false.
Then we have that $$\exists x : F(x)$$
Its negation is $$\forall x : \neg F(x)$$
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The negation of
"one of the sentences is false"
is
"no sentence is false or more than one sentences are false"
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To negate a sentence, we negate the primary operator. For example, to negate "$\exists x. f(x)=4$", we negate both the quantifier and the "=" operator but leave $f$ alone:
Original: $\exists x. f(x)=4$
Negated: $\forall x. f(x)\ne4$
The quantifier in your example is "one of", that is, a 'there exists' quantifier. The primary operator in the sentence is "is". To negate the sentence, negate the quantifier to get 'for all' and the operator to get "is not" (or "are not"):
Original: One of the sentences is false.
Negated: All of the sentences are not false.
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If we take "One of the sentences is false" as "There exists a sentence that is false", it can be written as:
$∃(x) \text{ false} (x)$
And its negation would be:
$¬∃(x) \text{ false} (x)$
From here, the $¬$ can be pushed inwards.
$∀(x)¬\text{ false}$
This is basically saying "All of the sentences are not false".
I'm taking "One of the sentences is false" to mean "There is a false sentence".
"One of the sentences is false" is negated to "It is not the case that one of the sentences is false": that is, "there are no sentences which are false".
"There exists a sentence $S$ such that $S$ is false" is negated to "For all sentences $S$, have $S$ is not false": i.e. "For all sentences $S$, $S$ is true".