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 dustbins are plastic items. No plastic item is stone. All woods are stones. Conclusions: I. No dustbin is wood. II. No plastic item is wood. III. Some woods may be dustbins.
Only conclusions I and II follow
Logical Reasoning Syllogism Solution
This problem requires us to analyze logical statements and determine which conclusions can be validly drawn from them. This type of question is known as a syllogism.
Understanding the Statements
We are given three statements, which we must assume are true:
- Statement 1: All dustbins are plastic items.
- Statement 2: No plastic item is stone.
- Statement 3: All woods are stones.
Let's represent these statements using basic set theory concepts for clarity:
- Statement 1 implies that the set of 'dustbins' is completely contained within the set of 'plastic items'.
- Statement 2 implies that the set of 'plastic items' and the set of 'stones' have no elements in common. They are mutually exclusive sets.
- Statement 3 implies that the set of 'woods' is completely contained within the set of 'stones'.
From these statements, we can deduce relationships between the categories:
- Since all dustbins are plastic items (Statement 1) and no plastic item is stone (Statement 2), it logically follows that no dustbin can be a stone.
- Since all woods are stones (Statement 3) and no plastic item is stone (Statement 2), it logically follows that no plastic item can be wood.
Analyzing the Conclusions
Now, let's examine each conclusion based on the statements and our deductions.
Conclusion I: No dustbin is wood.
Let's trace the relationship:
- Statement 1: Dustbins → Plastic Items (All Dustbins are Plastic Items)
- Statement 2: Plastic Items \(\cap\) Stone = \(\emptyset\) (No Plastic Item is Stone)
- Combining 1 & 2: Dustbins must also be \(\cap\) Stone = \(\emptyset\) (No Dustbin is Stone)
- Statement 3: Woods → Stones (All Woods are Stones)
- Since No Dustbin is Stone, and All Woods are Stones, it means Dustbins and Woods belong to categories that are separate from each other (Dustbins are not Stones, Woods are Stones). Therefore, no dustbin can be wood.
Conclusion I logically follows from the statements.
Conclusion II: No plastic item is wood.
Let's trace this relationship:
- Statement 2: Plastic Items \(\cap\) Stone = \(\emptyset\) (No Plastic Item is Stone)
- Statement 3: Woods → Stones (All Woods are Stones)
- Since All Woods are Stones, and No Plastic Item is Stone, there can be no overlap between Plastic Items and Woods. If plastic items were woods, they would have to be stones (because all woods are stones), but plastic items are not stones. This is a contradiction. Therefore, no plastic item is wood.
Conclusion II logically follows from the statements.
Conclusion III: Some woods may be dustbins.
This conclusion suggests a possibility of overlap between Woods and Dustbins.
- From our analysis of Conclusion I, we established that No Dustbin is Stone.
- From Statement 3, we know that All Woods are Stones.
- For 'Some woods to be dustbins' to be possible, there would need to be an intersection between the set of Woods and the set of Dustbins.
- However, Dustbins are not Stones, and Woods are Stones. These two sets (Dustbins and Woods) are fundamentally separated by their relationship (or lack thereof) with Stones.
- Therefore, there is no possibility, not even a 'may be', for some woods to be dustbins. The sets 'Woods' and 'Dustbins' are mutually exclusive based on the given statements.
Conclusion III does not follow from the statements; it is impossible according to the statements.
Summary of Conclusion Analysis
Based on our analysis:
- Conclusion I (No dustbin is wood) is valid.
- Conclusion II (No plastic item is wood) is valid.
- Conclusion III (Some woods may be dustbins) is invalid.
Therefore, only conclusions I and II follow from the given statements.
| Conclusion | Follows? | Reasoning |
|---|---|---|
| I. No dustbin is wood. | Yes | Dustbins are plastic items; Plastic items are not stones; Woods are stones. → Dustbins are not stones; Woods are stones. → No overlap between Dustbins and Woods. |
| II. No plastic item is wood. | Yes | Plastic items are not stones; Woods are stones. → No overlap between Plastic Items and Woods. |
| III. Some woods may be dustbins. | No | Dustbins are not stones; Woods are stones. → No overlap possible. |
Revision Table: Syllogism Basics
Understanding the types of statements in syllogism is key.
| Statement Type | Form | Example |
|---|---|---|
| Universal Affirmative | All A are B | All cats are mammals. |
| Universal Negative | No A is B | No dog is a cat. |
| Particular Affirmative | Some A are B | Some students are tall. |
| Particular Negative | Some A are not B | Some students are not tall. |
Additional Information: Syllogism Rules and Deductions
Syllogism is a form of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.
Common rules for determining validity involve the distribution of terms and the quality (affirmative/negative) and quantity (universal/particular) of the statements. For example:
- If both statements are particular, no universal conclusion can be drawn.
- If both statements are negative, no valid conclusion can be drawn.
- If one statement is negative, the conclusion must be negative.
- A term distributed in the conclusion must be distributed in the statements.
- The middle term (the term appearing in both statements but not the conclusion) must be distributed in at least one statement.
In this specific problem, the statements 'All dustbins are plastic items' and 'No plastic item is stone' lead to the conclusion 'No dustbin is stone' because 'plastic item' is the middle term distributed in the negative statement.
Similarly, 'No plastic item is stone' and 'All woods are stones' lead to 'No plastic item is wood' because 'stone' is the middle term distributed in the universal negative statement ('No plastic item is stone').
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 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.
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.
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.
