Analyzing the Negation of the Mango Statement

The question asks for the logically necessary true statement if the given statement "All the mangoes in the basket are good" is false.

Let the statement be represented using predicate logic. Let $M$ be the set of mangoes in the basket and $Good(m)$ be the predicate that mango $m$ is good.

The original statement is: $∀m \in M: Good(m)$ (This means for all mangoes $m$ in the basket $M$, the condition $Good(m)$ is true).

We are given that this statement is false. The negation of a universal statement ($∀$) is an existential statement ($∃$).

The negation of $∀m \in M: Good(m)$ is:

This is logically equivalent to:

In words, this means "There exists at least one mango $m$ in the basket $M$ such that the condition $Good(m)$ is false" (i.e., the mango is not good).

Evaluating the Options

Let's analyze the given options based on the derived negation ($∃m \in M: ¬Good(m)$):

• Option 1: All the mangoes in the basket are not good. Represents: $∀m \in M: ¬Good(m)$. This is stronger than necessary. If only one mango is not good, the original statement is false, but this option is not necessarily true.
• Option 2: No mango in the basket is good. Represents: $∀m \in M: ¬Good(m)$. Same as Option 1.
• Option 3: In the basket, some of the mangoes are good and some are not good. Represents: $(∃m \in M: Good(m)) ∧ (∃m \in M: ¬Good(m))$. This statement requires *both* good and not-good mangoes to exist. If all mangoes were not good, the original statement would be false, but this option would be false. Thus, it is not necessarily true.
• Option 4: There exists at least one mango in the basket that is not good. Represents: $∃m \in M: ¬Good(m)$. This is the direct logical negation of the original statement and must be true if the original statement is false.

Therefore, the statement that is necessarily true is Option 4.