Two statements are given, followed by two conclusions numbered I and II. Assuming the statements to be true, even if they seem to be at variance with commonly known facts, decide which of the conclusions logically follow(s) from the statements. Statements : Some apples are bananas. Some mangos are apples. Conclusions : (I) Some mangos are bananas. (II) Some bananas are apples.
Only conclusion II follows
Understanding the Logic Reasoning Problem
This question asks us to analyze two statements and determine which of the given conclusions logically follow from them. We must assume the statements are true, even if they contradict common knowledge. This type of problem is based on syllogism, a form of logical argument where a conclusion is inferred from two or more statements (premises).
Analyzing the Statements and Conclusions
Let's break down the given statements and conclusions:
Statements:
- Some apples are bananas.
- Some mangos are apples.
Conclusions:
- Some mangos are bananas.
- Some bananas are apples.
Analyzing Conclusion I: Some mangos are bananas.
We are told that some apples are bananas and some mangos are apples. Let's think about the relationship between mangos and bananas based on their connection to apples.
- Statement 1 establishes a partial overlap between the set of apples and the set of bananas.
- Statement 2 establishes a partial overlap between the set of mangos and the set of apples.
We can represent these relationships using diagrams (though we won't draw them explicitly here, we can think about them). The 'some' relationship means there is at least one element common to both sets. However, these statements do not give us any direct or indirect necessary connection between mangos and bananas. It is possible that the apples that are mangos are different from the apples that are bananas. Therefore, there is no guarantee that any mango is also a banana.
Conclusion I, "Some mangos are bananas," does not necessarily follow from the given statements. It might be true, but it is not logically compelled by the statements.
Analyzing Conclusion II: Some bananas are apples.
Let's look at the statements again, specifically Statement 1:
- Statement 1: Some apples are bananas.
This statement explicitly tells us that there is an overlap between the set of apples and the set of bananas. If some apples have the property of being bananas, this implies that there are some members of the set of bananas that have the property of being apples. In other words, the relationship "Some A are B" is symmetrical in terms of overlap – it implies "Some B are A".
Conclusion II, "Some bananas are apples," directly follows from Statement 1. If 'Some apples are bananas' is true, then 'Some bananas are apples' must also be true.
Determining Which Conclusions Follow
Based on our analysis:
- Conclusion I (Some mangos are bananas) does not logically follow from the statements.
- Conclusion II (Some bananas are apples) logically follows from Statement 1.
Therefore, only conclusion II follows from the given statements.
Summary of Conclusions
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I) Some mangos are bananas. | No | Statements provide no necessary link between mangos and bananas. |
| II) Some bananas are apples. | Yes | Directly implied by Statement 1 ("Some apples are bananas"). |
Revision Table: Syllogism Rules for "Some"
| Statement Type | Implication / Conversion |
|---|---|
| Some A are B | Implies Some B are A |
| Some A are B | Does not imply All A are B |
| Some A are B and Some B are C | Does not imply Some A are C |
Additional Information: Understanding Syllogisms
Syllogisms are a core part of deductive reasoning. They consist of premises (statements) and a conclusion. In these problems, the truth of the premises is assumed, and the task is to determine if the conclusion is necessarily true based *only* on those premises.
Keywords like "All", "No", and "Some" define the relationship between categories:
- All: Indicates complete inclusion (e.g., All dogs are mammals).
- No: Indicates complete exclusion (e.g., No fish are birds).
- Some: Indicates at least one, but not necessarily all. It signifies partial inclusion or overlap (e.g., Some students are athletes). "Some" is interpreted as "at least one and possibly all".
Understanding how these keywords interact in different statement combinations is key to solving syllogism problems. The statements "Some A are B" and "Some B are A" are equivalent in terms of the existence of an overlap.
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.
