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').
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.
