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 pencils are pens. All pens are papers. All pencils are colours. Conclusions : I. All colours are pens. II. Some colours are pencils. III. Some papers are pens.
Only conclusions II and III follow.
Analyzing Syllogism Statements and Conclusions
This problem requires us to analyze given statements and determine which of the provided conclusions logically follow. This type of problem is common in logical reasoning and is often solved using Venn diagrams or by understanding the relationships between categories.
Understanding the Statements
We are given three statements about the relationship between different items: pencils, pens, and papers.
- Statement 1: Some pencils are pens. This indicates an overlap between the set of pencils and the set of pens. There is at least one pencil that is also a pen, and vice versa.
- Statement 2: All pens are papers. This means that the set of pens is a subset of the set of papers. Every item that is a pen is also a paper.
- Statement 3: All pencils are colours. This means that the set of pencils is a subset of the set of colours. Every item that is a pencil is also a colour.
We must assume these statements are true, regardless of common knowledge.
Evaluating the Conclusions
Now let's examine each conclusion based on the truth of the statements.
Conclusion I: All colours are pens.
This conclusion states that the set of colours is a subset of the set of pens. Let's look at the statements:
- Statement 3 says, "All pencils are colours." This means pencils are inside colours.
- Statement 1 says, "Some pencils are pens." This means there's an overlap between pencils and pens.
If all pencils are colours, and some pencils are pens, it means that some colours are also pens (specifically, the colours that are pencils and also pens). However, Statement 3 does not say all colours are pencils; it only says all pencils are colours. It's possible for there to be colours that are not pencils. There is no statement linking all colours directly to pens in a way that suggests all colours must be pens. For example, there could be a colour (like red) that is not a pencil, and thus not necessarily a pen. Therefore, this conclusion does not logically follow from the statements.
Conclusion II: Some colours are pencils.
This conclusion states that there is an overlap between the set of colours and the set of pencils. Let's look at Statement 3:
- Statement 3 says, "All pencils are colours."
If every single pencil is also a colour, then it must be true that some part of the set of colours consists of pencils. This conclusion directly follows from Statement 3. If all members of set A are members of set B, then it is necessarily true that some members of set B are members of set A.
Conclusion III: Some papers are pens.
This conclusion states that there is an overlap between the set of papers and the set of pens. Let's look at Statement 2:
- Statement 2 says, "All pens are papers."
If every single pen is also a paper, then it must be true that some part of the set of papers consists of pens. This conclusion directly follows from Statement 2, similar to Conclusion II. If all members of set A are members of set B, then it is necessarily true that some members of set B are members of set A.
Summary of Conclusions
- Conclusion I (All colours are pens): Does not follow.
- Conclusion II (Some colours are pencils): Follows.
- Conclusion III (Some papers are pens): Follows.
Based on our analysis, only conclusions II and III logically follow from the given statements.
Revision Table: Syllogism Analysis
| Statement/Conclusion | Relationship Described | Follows? | Reasoning |
|---|---|---|---|
| Statement 1 | Some Pencils are Pens | Given | Partial overlap Pencils & Pens |
| Statement 2 | All Pens are Papers | Given | Pens is subset of Papers |
| Statement 3 | All Pencils are Colours | Given | Pencils is subset of Colours |
| Conclusion I | All Colours are Pens | No | Statements don't support Colours being subset of Pens. |
| Conclusion II | Some Colours are Pencils | Yes | Directly follows from Statement 3 (All Pencils are Colours). |
| Conclusion III | Some Papers are Pens | Yes | Directly follows from Statement 2 (All Pens are Papers). |
Additional Information: Syllogism and Logical Deduction
This problem is an example of a syllogism, a form of logical reasoning where a conclusion is drawn from two or more statements (premises). Understanding the basic types of categorical propositions (All A are B, No A are B, Some A are B, Some A are not B) is crucial.
- "All A are B" implies "Some B are A". (Converse is not always true)
- "Some A are B" implies "Some B are A". (Converse is true)
- "No A are B" implies "No B are A".
In this problem, "All pencils are colours" means the set of pencils is entirely contained within the set of colours. This necessarily means that within the set of colours, there are some items that are pencils. Similarly, "All pens are papers" means the set of pens is entirely within the set of papers, so some papers must be pens. These deductions are always valid.
Venn diagrams are often helpful tools for visualizing these relationships and testing conclusions. Drawing circles representing the sets and indicating overlap or containment can make the logical connections clearer.
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.
