Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements. Statement : . All computers are hard-drives. No hard-drive is a compact-disc. Conclusions: I. Some hard-drives are computers. II. No computer is a compact-disc.
Both conclusions I and II follow.
Understanding Syllogism and Logical Deduction
This question asks us to analyze given statements and determine which of the provided conclusions logically follow from them. This type of problem falls under the category of logical reasoning, specifically syllogisms. We must assume the statements are true, even if they contradict common knowledge, and use only the information provided in the statements to evaluate the conclusions.
Analyzing the Given Statements
We have two statements:
- All computers are hard-drives.
- No hard-drive is a compact-disc.
Let's represent these statements using a simplified notation or visualize them mentally using something like Venn diagrams.
- Statement 1: This means the set of 'computers' is entirely contained within the set of 'hard-drives'.
- Statement 2: This means the set of 'hard-drives' and the set of 'compact-discs' have no overlap whatsoever. They are mutually exclusive.
Evaluating the Conclusions
Now, let's examine each conclusion based on the statements.
Conclusion I: Some hard-drives are computers.
- Statement 1 says "All computers are hard-drives".
- If all members of set A are members of set B (All A are B), then it is always true that some members of set B are members of set A (Some B are A). For example, if all apples are fruits, then some fruits are apples.
- Applying this logic to Statement 1, if "All computers are hard-drives", it logically follows that "Some hard-drives are computers".
- Thus, Conclusion I logically follows from Statement 1.
Conclusion II: No computer is a compact-disc.
- Statement 1 tells us that computers are a part of hard-drives. (Computers <span>⊆</span> Hard-drives)
- Statement 2 tells us that hard-drives and compact-discs have no relationship. (Hard-drives <span>∩</span> Compact-discs <span>=</span> <span>∅</span>)
- Combining these two statements: If computers are inside the group of hard-drives, and the group of hard-drives has no overlap with compact-discs, then the computers (being inside the hard-drives group) can also have no overlap with the compact-discs.
- Visually: If the 'Computers' circle is inside the 'Hard-drives' circle, and the 'Hard-drives' circle is completely separate from the 'Compact-discs' circle, then the 'Computers' circle must also be completely separate from the 'Compact-discs' circle.
- Therefore, "No computer is a compact-disc" logically follows from the combination of Statement 1 and Statement 2.
Conclusion Summary
Based on our analysis:
- Conclusion I ("Some hard-drives are computers") logically follows from Statement 1.
- Conclusion II ("No computer is a compact-disc") logically follows from Statement 1 and Statement 2.
Since both conclusions logically follow from the given statements, the correct option is the one stating that both conclusions I and II follow.
| Statement/Conclusion | Analysis | Logical Follows? |
|---|---|---|
| Statement 1: All computers are hard-drives. | Computers are a subset of Hard-drives. | N/A |
| Statement 2: No hard-drive is a compact-disc. | Hard-drives and Compact-discs are separate. | N/A |
| Conclusion I: Some hard-drives are computers. | Inverse of "All A are B" implies "Some B are A". | Yes |
| Conclusion II: No computer is a compact-disc. | If A <span>⊆</span> B and B <span>∩</span> C <span>=</span> <span>∅</span>, then A <span>∩</span> C <span>=</span> <span>∅</span>. | Yes |
Revision Table: Key Syllogism Types
| Type | Statement Form | Example |
|---|---|---|
| A (Universal Affirmative) | All S are P. | All cats are mammals. |
| E (Universal Negative) | No S is P. | No dog is a cat. |
| I (Particular Affirmative) | Some S are P. | Some students are smart. |
| O (Particular Negative) | Some S are not P. | Some students are not smart. |
Additional Information: Solving Syllogism Problems
When solving syllogism problems, remember these key points:
- Assume statements are true: Treat the given statements as facts, regardless of real-world accuracy.
- Analyze each statement: Understand the relationship between the terms in each statement (e.g., inclusion, exclusion, partial overlap).
- Combine statements: Look for a common term (the middle term) that links the two statements. In this case, 'hard-drives' is the middle term.
- Evaluate conclusions: Check if each conclusion *must* be true based *only* on the information in the statements. If there's any possibility, however remote, that the conclusion is false, it does not logically follow.
- Use diagrams: Visual tools like Venn diagrams can be very helpful in representing the relationships described in the statements and testing the validity of the conclusions.
- Basic Inferences: Remember that "All A are B" implies "Some B are A". "No A is B" implies "No B is A" and "Some A are not B", and "Some B are not A".
Mastering the analysis of statements and conclusions in logical reasoning problems is crucial for many competitive exams.
Important Questions from Syllogism
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
Some cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is true, even if it appears to be at variance with commonly known facts, decide which of the given conclusions logically follow(s) from the statements.
Statements:
All employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
