If the above statement is false, then which one of the following statements is necessarily true?
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:
$¬(∀m \in M: Good(m))$This is logically equivalent to:
$∃m \in M: ¬Good(m)$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.
Important Questions from Logical Deduction
- The following observation is made about the scores obtained by 100 students in an exam:
'For each student, there exists another student in the class such that their scores are at most ten marks away.'
If the above statement is false, which one of the following statements is necessarily true? - Consider the following two phonological rules:
Rule 1: Vowel Epenthesis of [ɪ] after sibilant-ending stems
Rule 2: Progressive Voicing Assimilation of the plural affix
Which ONE of the following rule ordering relations apply in the case of regular English pluralization as in ‘horse’ – ‘horses’? Consider a linear arrangement of seven bulbs, each of which can be in the ON or OFF states. The initial configuration of the bulbs is shown in the figure. In every Step, the states of the bulbs are changed based on the following rules:
- Any OFF bulb with exactly one ON neighbor at the end of the previous Step is turned ON.
- Any ON bulb with both neighbors ON at the end of the previous Step is turned OFF.
- The state of any bulb not meeting the conditions above is left unchanged.
The state of bulbs at the end of Step 1 and Step 2 are also shown in the figure.
The number of bulbs which are ON at the end of Step 8 is ______
- Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you", Ramya told Josephine. Consider a five-digit number PQRST that has distinct digits P, Q, R, S and T, and satisfies the following conditions:
$P < Q$
$S > P > T$
$R < T$
If integers 1 through 5 are used to construct such a number, the value of P is:
