Consider the following three statements:
(i) Some roses are red.
(ii) All red flowers fade quickly.
(iii) Some roses fade quickly.
Which of the following statements can be logically inferred from the above statements?
Logical Inference Analysis
This question requires analyzing the logical connection between three statements to determine when the third statement can be definitively inferred.
Statement Breakdown
- Statement (i): Some roses are red. This is a particular affirmative statement. It means there exists at least one rose that is red. Let R be the set of roses and D be the set of red things. Then, $R \cap D \neq \emptyset$.
- Statement (ii): All red flowers fade quickly. This is a universal affirmative statement. Assuming 'red' implies 'red flower' in this context, it means everything that is red also fades quickly. Let F be the set of things that fade quickly. Then, $D \subseteq F$.
- Statement (iii): Some roses fade quickly. This is a particular affirmative statement, suggesting $R \cap F \neq \emptyset$.
Deriving the Conclusion
We need to find the condition under which statement (iii) is a logical consequence of statements (i) and (ii).
- Assume statements (i) and (ii) are true.
- From statement (i), we know there is at least one entity that is both a rose and red. Let's call this entity 'x'. So, $x$ is a rose AND $x$ is red.
- From statement (ii), we know that if something is red, it fades quickly.
- Since 'x' is red (from step 2), it must also fade quickly (based on step 3).
- Combining these facts, 'x' is a rose (from step 2) and 'x' fades quickly (from step 4).
- This directly implies that there exists at least one rose that fades quickly, which is statement (iii).
Therefore, if statements (i) and (ii) are true, statement (iii) must logically follow. This corresponds to the condition described in Option 3.
Evaluating Other Options
The validity of statement (iii) depends entirely on the truthfulness of statements (i) and (ii). Scenarios where either (i) or (ii) (or both) are false do not impact the logical inference derived from them being true. Option 3 correctly captures the condition under which the inference holds.
Important Questions from Syllogism
Statements:
1. All heroes are winners.
2. All winners are lucky people.
Inferences:
I. All lucky people are heroes.
II. Some lucky people are heroes.
III. Some winners are heroes.
Which of the above inferences can be logically deduced from statements 1 and 2?Human beings are one among many creatures that inhabit an imagined world. In this imagined world, some creatures are cruel. If in this imagined world, it is given that the statement “Some human beings are not cruel creatures” is FALSE, then which of the following set of statement(s) can be logically inferred with certainty?
(i) All human beings are cruel creatures.
(ii) Some human beings are cruel creatures.
(iii) Some creatures that are cruel are human beings.
(iv) No human beings are cruel creatures.- Given below are four statements.
Statement 1: All students are inquisitive.
Statement 2: Some students are inquisitive.
Statement 3: No student is inquisitive.
Statement 4: Some students are not inquisitive.
From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student in the class. - Based only on the truth of the statement ‘Some humans are intelligent', which one of the following options can be logically inferred with certainty?
Given below are two statements and four conclusions drawn based on the statements.
Statement 1: Some bottles are cups.
Statement 2: All cups are knives.
Conclusion I: Some bottles are knives.
Conclusion II: Some knives are cups.
Conclusion III: All cups are bottles.
Conclusion IV: All knives are cups.
Which one of the following options can be logically inferred?
