‘All the mangoes in the basket are good.’

If the above statement is false, then which one of the following is necessarily true?

The correct option is: There exists at least one mango in the basket that is not good.

Step-by-Step Explanation:

1. Understanding the Statement:
The given statement is: "All the mangoes in the basket are good."
In mathematical logic, this is a universal statement (quantified by "all" or "every"). If we define:
- $M\left(x\right)$ as "$x$ is a mango in the basket"
- $G\left(x\right)$ as "$x$ is good"
Then the statement can be represented symbolically as:

$\forall x\left(M\right(x)\to G(x\left)\right)$

2. Negating the Statement:
We are told that this statement is false. If a statement is false, its negation must be true.
The negation of a universal statement ($\forall$) is an existential statement ($\exists$). The rule of negation states that the negation of "All A are B" is "Some A are not B" (or "There exists at least one A that is not B").
Symbolically, the negation is:

$\neg [\forall x(M\left(x\right)\to G\left(x\right)\left)\right]\equiv \exists x\left(M\right(x)\wedge \neg G(x\left)\right)$

3. Translating Back to English:
The logical form $\exists x\left(M\right(x)\wedge \neg G(x\left)\right)$ translates to:
"There exists at least one mango in the basket that is not good."

4. Analyzing Other Options:
- "No mango in the basket is good" (or "All the mangoes in the basket are not good") is not necessarily true. For the original statement to be false, we do not need every single mango to be bad; having just one bad mango is sufficient to disprove "all mangoes are good".
- "In the basket, some of the mangoes are good & some are not good" is also not necessarily true because it is possible that all mangoes are bad, which would still satisfy the condition of the statement being false.

Therefore, the only statement that is necessarily true when the original statement is false is that there is at least one mango in the basket that is not good.