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.
Important Questions from Conventional 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:
No bank is an office.
All offices are stalls.
Conclusions:
I. No bank is a stall.
II. No stall is a bank.
III. Some stalls are offices.
IV. All the stalls are offices
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 flowers are beautiful.
Vaidehi is beautiful.
Conclusions:
I. Vaidehi is a flower.
II. Some beautiful are flowers.
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
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 fingers are toes.
Some toes are rings.
Some rings are hands.
Conclusions:
I. Some hands are toes.
II. Some rings are fingers.
III. Some hands are fingers.
V. Some fingers are rings.
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 polygons are angles.
All angles are diagonals.
All cones are cubes.
All cubes are decagons.
No diagonal is a cube.
Conclusions:
I. Some diagonals are polygons.
II. All diagonals are decagons.
III. No polygon is a cone.
IV. Some cubes are angles.
