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. All M are A. II. No A is R. Conclusion: I. No M is R. II. Some A are M. III. Some R are A.
Both conclusions I and II follows
Understanding Statements and Conclusions in Logical Reasoning
This question asks us to analyze given statements and determine which of the provided conclusions logically follow from them. This type of problem falls under the category of logical reasoning, specifically dealing with syllogisms.
Analyzing the Given Statements
We are provided with two statements:
- Statement I: All M are A.
- Statement II: No A is R.
We must assume these statements are true, even if they contradict common knowledge.
Analyzing the Given Conclusions
We need to evaluate three conclusions:
- Conclusion I: No M is R.
- Conclusion II: Some A are M.
- Conclusion III: Some R are A.
Evaluating Each Conclusion Based on Statements
Let's examine each conclusion to see if it logically follows from the given statements. We can visualize the relationships using Venn diagrams or apply rules of logic.
Evaluation of Conclusion I: No M is R
Statement I tells us that the entire set of 'M' is contained within the set of 'A'. Statement II tells us that the set of 'A' and the set of 'R' have absolutely no overlap.
If all 'M' are inside 'A', and 'A' has nothing in common with 'R', then it must be true that 'M' also has nothing in common with 'R'. Therefore, "No M is R" logically follows from the given statements.
Evaluation of Conclusion II: Some A are M
Statement I says "All M are A". This means that every single element of M is also an element of A. If there are any M's (which is the standard assumption in these types of problems unless specified otherwise), then those M's are also A's. This implies that the group of A's includes the group of M's. Thus, there are some elements within the set A that are also elements of the set M. Therefore, "Some A are M" logically follows from "All M are A".
Evaluation of Conclusion III: Some R are A
Statement II says "No A is R". This means that there is no overlap between the set of 'A' and the set of 'R'. If no A is R, it also means that no R is A. The conclusion "Some R are A" suggests that there is at least some overlap between R and A, which directly contradicts Statement II. Therefore, "Some R are A" does not logically follow from the given statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I (No M is R): Follows.
- Conclusion II (Some A are M): Follows.
- Conclusion III (Some R are A): Does not follow.
Thus, both Conclusion I and Conclusion II follow from the given statements.
Final Answer
The conclusions that follow are I and II.
| Conclusion | Logically Follows? | Reasoning |
|---|---|---|
| I. No M is R | Yes | If M is subset of A and A & R are disjoint, then M & R are disjoint. |
| II. Some A are M | Yes | "All M are A" implies some A are M. |
| III. Some R are A | No | "No A is R" means A and R are disjoint, contradicting "Some R are A". |
Revision Table: Logical Reasoning Basics
| Statement Type | Representation | Key Implication (if valid) |
|---|---|---|
| All X are Y | X is a subset of Y | Some Y are X (usually assumed if X is non-empty) |
| No X is Y | X and Y are disjoint sets | No Y is X |
| Some X are Y | X and Y have overlap | Some Y are X |
| Some X are not Y | Part of X is outside Y | No direct implication about Y and X relation |
Additional Information: Syllogism and Venn Diagrams
Syllogism is a form of logical reasoning where a conclusion is drawn from two given or assumed propositions (premises). In this problem, the statements are the premises.
Venn diagrams are often used to visually represent the relationships between sets described in the statements. For example:
- "All M are A" can be shown as a circle for M completely inside a circle for A.
- "No A is R" can be shown as a circle for A and a circle for R with no overlapping area between them.
By drawing these relationships together, one can visually check if the conclusions hold true. In this case:
- Draw circle M inside circle A.
- Draw circle R separate from circle A.
Looking at the diagram:
- Is there any overlap between M and R? No, because M is inside A, and A doesn't overlap with R. This supports Conclusion I.
- Is there any overlap between A and M? Yes, the entire M circle is inside A, meaning the part of A that is M is the M circle itself. This supports Conclusion II.
- Is there any overlap between R and A? No, they are separate according to Statement II. This contradicts Conclusion III.
This visual method confirms our logical deduction.
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:
1. All rugs are blankets.
2. All blankets are pillows.
3. Some blankets are frames.
Conclusions:
I. All pillows are rugs.
II. Some pillows are rugs.
III. All rugs are frames
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. All T are G.
II. All G are H.
Conclusions:
I. All T are H.
II. No G is T.
III. All H are G.
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 reds are pinks.
All whites are pinks.
All pinks are oranges.
Conclusions:
I. All whites are oranges.
II. All oranges are reds.
III. Some oranges are whites.
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. All R are M.
II. All L are M.
Conclusions:
I. Some M are not R.
II. Some L are R.
III. Some M are L.
Three statements are given, followed by two conclusions numbered I and II. 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 c are f.
No f is t.
All p are c.
Conclusions:
I. Some t are c.
II. No p is t.
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 bottles are towels.
No towel is a pillow.
All bottles are coats.
Conclusions:
I. Some coats are towels.
II. No coat is a towel.
III. Some bottles are pillows.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 Tim are Jack.
No Jack is Ben.
Some Ben is Ruth.
Conclusions:
I. No Ruth is Jack.
II. Some Ben are Jack.
III. No Ben is Tim.
In this question, 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 follows/follow from the statements.
Statements:
Some roses are yellow.
Some yellow are buses.
All buses are green.
Conclusions:
I. All yellow are roses.
II. Some buses are yellow.
III. All yellow are green.
- Sunita travels 6 Km towards north east. Then travels 9 Km towards west. From there, she goes 6 Km south-west and finally goes 3 Km towards east. How far is she from her initial position?
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 laptops are phones.
Some phones are capsules.
Some capsules are pens.
Conclusions:
I. Some laptops are capsules.
II. Some phones are pens.
III. Some pens are capsules.
IV. Some pens are phones.
Important Questions from Conventional Syllogism
In this question, three statements followed by two conclusions numbered I and II have been given. You have to take the given statements to be true even if they seem to be at variance from the commonly facts and then decide which of the given conclusions logically follows from the given statements disregarding commonly known facts.
Statements: Some flats are apartments.
No apartment is a hall.
Some halls are rooms.
Conclusions: I. At least some rooms are flats.
II. No apartment is a room.
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:
I. Some blue are red.
II. Some green are red.
Conclusions:
I. No blue is green.
II. No red is green.
Given below are two statements, Consider these statements to be true even if they seem factuality absurd, Read the conclusions and then decide which of the given conclusions logically follow(s) from the given statements?
Statements :
1. All vases are flowers.
2. No flowers is a plant.
Conclusions :
1. No vases is a plant.
2. Some plant are vases
In the question two statements are given, followed by three conclusions, I, II and III. You have to consider the statements to be true even if it seems to be at variance from commonly known facts. You have to decide which of the given conclusions, if any, follows from the given statements.
Statement 1 : Some cars are scooters.
Statement 2 : All scooters are buses.
Conclusion I : Some scooters are cars.
Conclusion II : Some buses are cars.
Conclusion III : All cars are buses.
In the question below are given three statements followed by two conclusions. You have to take each of the given statements to be true. Read the conclusions and then decide which of the conclusions can be logically derived.
Statements:
Some schools are colleges.
No college is university.
All universities are hospitals.
Conclusions:
I. Some schools are universities.
II. At least some hospitals are colleges.
