Three statements are given, followed by three conclusions numbered I, II and III.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: No cup is a plate. All cups are utensils. Some bottles are cups. Conclusions: I. Some utensils are not plates. II. Some bottles are not plates. III. No bottle is a plate.
Only conclusions I and II follow
Analyzing Statements and Conclusions in Logical Reasoning
This problem requires us to analyze a set of statements and determine which of the given conclusions logically follow from them, assuming the statements are true. This type of question tests our ability to apply logical deduction principles, often visualized using methods like Venn diagrams.
Understanding the Given Statements
We have three statements:
- No cup is a plate.
- All cups are utensils.
- Some bottles are cups.
Let's break down what each statement tells us about the relationships between the categories: Cups, Plates, Utensils, and Bottles.
- Statement 1 establishes a complete separation between the category of 'cups' and the category of 'plates'. There is absolutely no overlap between them.
- Statement 2 says that the category of 'cups' is entirely contained within the category of 'utensils'. Every single cup is also a utensil.
- Statement 3 indicates that there is an overlap between the category of 'bottles' and the category of 'cups'. Some, but not necessarily all, bottles are also cups.
Combining the Statements
Now, let's see how these statements connect:
- From Statement 3 ("Some bottles are cups") and Statement 2 ("All cups are utensils"), we know that the 'some bottles' that are cups must also be utensils. So, some bottles are utensils.
- From Statement 3 ("Some bottles are cups") and Statement 1 ("No cup is a plate"), we know that the 'some bottles' that are cups cannot be plates, because no cup can be a plate. So, some bottles are not plates.
- From Statement 2 ("All cups are utensils") and Statement 1 ("No cup is a plate"), we know that the entire group of 'cups' (which is inside 'utensils') has no overlap with 'plates'. Since cups are part of utensils, the portion of utensils that consists of cups is not plates. So, some utensils (specifically, the cups) are not plates.
Evaluating the Conclusions
We now evaluate each conclusion based on the combined information from the statements.
Conclusion I: Some utensils are not plates.
- From our analysis, we saw that 'All cups are utensils' and 'No cup is a plate'.
- This means the category 'Cups' is within 'Utensils', and 'Cups' have no relationship with 'Plates'.
- Therefore, the items that are cups (and are also utensils) are not plates. This confirms that some utensils are not plates (at least the ones that are cups).
- Conclusion I logically follows.
Conclusion II: Some bottles are not plates.
- From our analysis, we know that 'Some bottles are cups' and 'No cup is a plate'.
- The bottles that fall into the 'cups' category cannot be 'plates' because no cup is a plate.
- Since 'some bottles' are cups, it means those specific bottles are not plates.
- Therefore, some bottles are not plates.
- Conclusion II logically follows.
Conclusion III: No bottle is a plate.
- We know 'Some bottles are cups' and 'No cup is a plate'. This tells us the bottles that are cups are not plates.
- However, the statements do not give us any information about the bottles that are not cups.
- It is possible that some bottles exist that are not cups, and these non-cup bottles might be plates. The statements do not rule this out.
- Since we cannot definitively say that no bottle is a plate based on the given statements, this conclusion does not logically follow.
- Conclusion III does not logically follow.
Summary of Conclusions
Based on our evaluation:
| Conclusion | Follows? | Reasoning |
|---|---|---|
| I. Some utensils are not plates. | Yes | Cups are utensils, and no cups are plates. Thus, some utensils (cups) are not plates. |
| II. Some bottles are not plates. | Yes | Some bottles are cups, and no cups are plates. Thus, some bottles (those that are cups) are not plates. |
| III. No bottle is a plate. | No | Statements don't provide information about bottles that are NOT cups. They could be plates. |
Therefore, only conclusions I and II logically follow from the given statements.
Revision Table: Logical Reasoning Concepts
| Statement Type | Relationship | Keywords |
|---|---|---|
| All A are B | A is a subset of B | Universal affirmative, inclusion |
| No A is B | A and B are disjoint sets | Universal negative, exclusion |
| Some A are B | A and B have overlap | Particular affirmative, intersection |
| Some A are not B | A and B have partial exclusion | Particular negative, partial exclusion |
Additional Information: Understanding Syllogisms
This question is based on the principles of syllogisms, a form of logical reasoning where a conclusion is derived from two or more premises (statements). Syllogisms involve quantifiers like "all", "some", and "no". To solve these problems effectively, it is helpful to:
- Identify the terms involved (e.g., Cup, Plate, Utensil, Bottle).
- Represent the relationships between these terms using diagrams (like Venn diagrams) or symbolic logic.
- Trace the flow of relationships from the statements to see if they support the conclusions.
- Pay close attention to the quantifiers ("some", "all", "no") as they define the extent of the relationship. "Some" means "at least one", and it does not imply "not all". "No" implies complete absence of overlap.
- Assume the statements are true, even if they contradict real-world knowledge. The task is to find what *logically follows* from the given premises alone.
Similar Questions
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.
Three statements are given followed by three conclusions numbered I, II and III. 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 spices are medicines.
All medicines are chemicals.
No chemical is cheap.
Conclusions:
I. Some medicines are cheap.
II. Some chemicals are spices.
III. No medicine is cheap.
Three statements are given followed by three conclusions numbered I, II and III. 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:
All oranges are apples.
Some oranges are bananas.
All bananas are fruits.
Conclusions:
I. Some fruits are apples.
II. Some apples are bananas.
III. Some oranges are fruits.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. Some Z are X.
II. Some W are Z.
Conclusions:
I. Some X are not W.
II. Some X are not Z.
Three Statements are given followed by Three conclusions numbered I, II and III. 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 mobiles are flat.
All flats are TVs.
Some TVs are plastic.
Conclusions:
I. Some mobiles are TVs.
II. Some flats are plastic.
III. All plastic are TVs.
Three statements are followed by three conclusions numbered I, II and III. You have to consider these statements to be true, even if they seem to be at variance with commonly known facts. Decide which of the given conclusions logically follow/s from the given statements.
Statements:
Some watches are projectors.
All projectors are lamps.
Some heaters are watches.
Conclusions:
(I) Some heaters are projectors.
(II) Some lamps are heaters.
(III) All watches are lamps
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. No A is B.
II. No C is B.
Conclusion:
I. Some C are not A.
II. Some A are B.
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 benches are tables.
Some tables are chairs.
Some chairs are pencils.
Conclusions:
I. Some benches are chairs.
II. Some tables are pencils.
III. Some pencils are chairs.
IV. Some pencils are tables.
In the following question below are given some statements followed by some conclusions based on those statements. Taking the given statements to be true even if they seem to be at variance from commonly known facts. Read all the conclusions and then decide which of the given conclusion logically follows the given statements.
Statements:
I. No tyre are white.
II. All grey are white.
Conclusions:
I. No white is tyre.
II. No tyre is grey.
III. Some white are not grey.
Important Questions from Syllogism
The statements below are followed by conclusions labeled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All twenty are thirty.
All thirty are forty.
All forty are sixty.
All sixty are seventy.
Conclusions:
I. Some forty are thirty.
II. Some seventy are sixty.
III. No thirty is twenty.The statements below are followed by two conclusions labelled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.
Statements:
All Strong are animals.
Some animals are Tigers.
All Tigers are Sharp.
Conclusions:
I. Some Strong are Sharp.
II. No Strong is Sharp.The statements below are followed by conclusions labelled I, II and III. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some women are weak.
Some weaks are female.
All female are iron.
All iron are gold.Conclusions:
I. Some weaks are iron.
II. Some gold are weaks.
III. Some women are female.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally established facts, decide which conclusion(s) logically and definitely follow(s) from the information given in the statements.Statements:
Some teaspoons are glasses.
All teddies are teaspoons.
Conclusions:
I. Some teddies are glasses.
II. Some glasses are teddies.
The statements below are followed by two conclusions labeled I and II. Assuming that the information in the statements is true, even if it appears to be at variance with generally Established facts, decide which conclusion(s) logically and definitely follow(s) from the Information, Given in the Statements.
Statements:
All bangles are rings.
Some rings are toys.
Some toys are dolls.
Conclusions:
I. Some rings are bangles.
II. Some dolls are rings.
