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 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.
Consider the given statement(s) to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statement 1: All elephants are men.
Statement 2: All men are socks.
Conclusion I: Some socks are elephants.
Conclusion II: All elephants are socks.In the question, a word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the question are represented by two classes of alphabets in two matrices given below. The columns and rows of Matrix 1 are numbered from 0 to 4 and that of Matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., 'A' can be represented by 04, 12, etc., and 'U' can be represented by 59, 66, etc. Similarly, you have to identify the set for the word 'ROOT'.
MATRIX I
0 1 2 3 4 0 B C D E A 1 D E A B C 2 C D E A B 3 A B C D E 4 E A B C D MATRIX II
5 6 7 8 9 5 R O N T U 6 T U R O N 7 O N T U R 8 U R O N T 9 N T U R O Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statements: 1. All fish are tortoise.
2. No tortoise is a crocodile.
Conclusion: 1. No crocodile is a fish.
II. No fish is a crocodile.
Consider the given statements to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statements: 1. All books are novels.
2. Some novels are poems.
Conclusions: I. Some books are poems.
II. Some poems are novels.Consider the statement to be true and decide which of the conclusions logically follow:
Statement:
All dogs are animals.
All animals are living beings.
Conclusion
I:Dog is not an animal.
II. Dog is an animal.In the question, a word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the matrices are represented by two classes of alphabets as in two matrices given below. The columns and rows of Matrix I are numbered from 0 to 4 and that of Matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., 'A' can be represented by 01, 13, etc., and 'S' can be represented by 55, 66, etc. Similarly, you have to identify the set for the word 'BOTH'.
MATRIX I
0 1 2 3 4 0 F A N O I 1 I O F A N 2 A N O I F 3 O F I N A 4 N I A F O MATRIX II
5 6 7 8 9 5 S E H B T 6 H S E T B 7 B T S E H 8 E H T B S 9 T S E H B Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statement I: All astronomers are scientists.
Statement II: Some scientists are shopkeepers.
Conclusion I: All astronomers are shopkeepers.
Conclusion II: All astronomers are scientists.
Conclusion III: Some shopkeepers are astronomers.
Conclusion IV: All scientists are astronomers.A word is represented by only one set of numbers as given in any one of the options. The sets of numbers are formed using two matrices below. The rows and columns of Matrix I are numbered 0 to 4 and Matrix II are numbered 5 to 9. A letter is represented by its row and column numbers. For example, ‘G’ can be coded as 12 or 21, and ‘L’ as 76 or 67.
Find the correct code for the word ‘LIFE’.
Consider the given statement/s to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement:
Statements: Some cars are caves.
Conclusion: I. Some cars are not buses
II. Some caves are buses
III. No cave is a bus
IV. Some caves are cars
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.
