Two 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 paintings are calligraphies. All sketching are calligraphies . Conclusions: I. At least some calligraphies are paintings. II. No sketching is a painting.
Only conclusion I follows.
Let's analyze the given statements and conclusions using principles of logical reasoning, often represented with Venn diagrams or set theory concepts. The goal is to determine which conclusions logically follow from the given statements, assuming the statements are true.
Statements Analysis and Interpretation
We have two statements:
- Statement 1: All paintings are calligraphies.
- Statement 2: All sketching are calligraphies.
Let's represent these statements using sets:
- Let P be the set of Paintings.
- Let C be the set of Calligraphies.
- Let S be the set of Sketching.
Statement 1 means that the set of Paintings (P) is a subset of the set of Calligraphies (C). In set notation, this is \(P \subseteq C\). This implies that every element in the set P is also an element in the set C.
Statement 2 means that the set of Sketching (S) is a subset of the set of Calligraphies (C). In set notation, this is \(S \subseteq C\). This implies that every element in the set S is also an element in the set C.
Visually, we can imagine a large circle representing Calligraphies. Inside this circle are smaller circles representing Paintings and Sketching. The statements tell us that Paintings are inside Calligraphies, and Sketching are inside Calligraphies. However, the statements do not give us any direct information about the relationship between Paintings and Sketching. They could be separate, overlapping, or one could be inside the other, as long as both are inside Calligraphies.
Conclusion I Analysis: At least some calligraphies are paintings.
This conclusion states that there is an overlap between the set of Calligraphies and the set of Paintings, specifically that the intersection is not empty and contains at least some elements.
From Statement 1, we know: All paintings are calligraphies (\(P \subseteq C\)).
If all elements of set P are also elements of set C, then the elements that are in P must also be in C. These elements are calligraphies that are also paintings. Therefore, the set of paintings (\(P\)) is exactly the set of 'calligraphies that are paintings'.
If the set of paintings is not empty (which is a standard assumption in these types of problems unless stated otherwise), then there exist items that are paintings. Since all paintings are calligraphies, these items are also calligraphies. Thus, there are items that are both calligraphies and paintings. This means 'some calligraphies are paintings' is true.
Alternatively, if \(P \subseteq C\), then any element in P is also in C. This means that the intersection of C and P (\(C \cap P\)) is equal to P. If P is not empty, then \(C \cap P\) is not empty, which implies that there exist elements common to both sets. These elements are calligraphies that are paintings.
Therefore, Conclusion I logically follows from Statement 1.
Conclusion II Analysis: No sketching is a painting.
This conclusion states that there is no overlap between the set of Sketching and the set of Paintings. In set notation, this means \(S \cap P = \emptyset\).
From the statements, we know:
- All paintings are calligraphies (\(P \subseteq C\)).
- All sketching are calligraphies (\(S \subseteq C\)).
Both sets P and S are subsets of C. However, the statements provide no direct information about the relationship between P and S. We can imagine different scenarios consistent with the statements:
- Sketching and Paintings are entirely separate subsets within Calligraphies. (e.g., S and P do not overlap) - In this case, "No sketching is a painting" would be true.
- Sketching and Paintings overlap within Calligraphies. (e.g., S and P share some common elements) - In this case, "Some sketching are paintings" would be true, and "No sketching is a painting" would be false.
- Paintings are a subset of Sketching within Calligraphies. (e.g., \(P \subseteq S\)) - In this case, "All paintings are sketching" would be true, and "No sketching is a painting" would be false (unless P is empty).
- Sketching are a subset of Paintings within Calligraphies. (e.g., \(S \subseteq P\)) - In this case, "All sketching are paintings" would be true, and "No sketching is a painting" would be false (unless S is empty).
Since there are possible scenarios (like overlap) where Conclusion II is false, Conclusion II does not logically follow from the statements. A conclusion only follows if it is true in *all* possible scenarios consistent with the statements.
Summary of Conclusions
- Conclusion I: At least some calligraphies are paintings. (Follows)
- Conclusion II: No sketching is a painting. (Does not follow)
Determining the Final Answer
Based on our analysis, only Conclusion I logically follows from the given statements.
| Statement/Conclusion | Relationship | Follows? |
|---|---|---|
| Statement 1 | All Paintings are Calligraphies (\(P \subseteq C\)) | Given |
| Statement 2 | All Sketching are Calligraphies (\(S \subseteq C\)) | Given |
| Conclusion I | At least some Calligraphies are Paintings (\(C \cap P \neq \emptyset\)) | Yes (derived from Statement 1) |
| Conclusion II | No Sketching is a Painting (\(S \cap P = \emptyset\)) | No (not necessarily true based on Statements 1 & 2) |
Revision Table: Key Concepts in Syllogisms
| Term | Explanation | Relevance Here |
|---|---|---|
| Syllogism | A form of deductive reasoning where a conclusion is drawn from two given or assumed propositions (premises). | This problem is a type of syllogism involving categorical propositions. |
| Categorical Proposition | A proposition that relates two categories (or terms). Types include "All A are B", "No A is B", "Some A are B", "Some A are not B". | The given statements and conclusions are categorical propositions. |
| Venn Diagrams | Diagrams using overlapping circles to visually represent the relationships between sets. | Useful tool to check the validity of conclusions based on statements. |
| Subset (\(\subseteq\)) | A set A is a subset of set B if every element of A is also an element of B. | "All Paintings are Calligraphies" means Paintings are a subset of Calligraphies. |
| Intersection (\(\cap\)) | The set of elements that are in both set A and set B. | "Some Calligraphies are Paintings" means the intersection of Calligraphies and Paintings is not empty. |
Additional Information: Understanding "Some" and "At Least Some"
In logic, the term "some" means "at least one". So, "Some calligraphies are paintings" is equivalent to "At least some calligraphies are paintings". This means there exists at least one item that belongs to both the category of calligraphies and the category of paintings.
When a statement says "All A are B", it automatically implies "Some B are A", provided that set A is not empty. This is because if every A is a B, then the collection of all A's is a part of B. The elements of A are B's that are also A's. If there are A's, then there are B's (specifically, the A's) that are also A's. If 'Paintings' is not an empty category, then 'All paintings are calligraphies' directly leads to the conclusion 'Some calligraphies are paintings'.
However, from "All A are B" and "All C are B", we cannot conclude anything definite about the relationship between A and C without further information. A and C could be disjoint, overlapping, or one could be a subset of the other, as long as they both remain subsets of B.
Similar Questions
Two 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 teachers are engineers.
All clerks are teachers.
Conclusions:
I. Some engineers are clerks.
II. Some teachers are clerks.
III. Some clerks are not engineers.
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 file is a folder.
Some files are PDFs.
All files are Excel sheets.
Conclusions:
I. Some PDFs are folders.
II. Some Excel sheets are folders.
III. Some PDFs are Excel sheets.
In this question three statements are given followed by three conclusions I, II and III. You have to consider the three 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(s) logically follow(s) from the given statements.
Statements:
All photocopiers are computers.
Some computers are laptops.
No laptop is a mobile.
Conclusions:
(I) No computer is a mobile.
(II) Some computers are photocopiers.
(III) All mobiles are photocopiers.
Two 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:
Some vehicles are helicopters.
All helicopters are cars.
Conclusions:
I. All vehicles are cars.
II. Some vehicles are cars.
Given below are two statements and two conclusions. Take the statements to be true even if they are at variance with commonly known facts, and decide whether the conclusion(s) follow(s) the given statements.
Statements:
All eyes are noses.
Some noses are lips.
Conclusions:
I. Some noses are eyes.
II. All lips are noses.
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 telescopes are periscopes.
Some telescopes are microscopes.
Some telescopes are kaleidoscopes.
Conclusions:
I. Some kaleidoscopes are telescopes.
II. Some microscopes are kaleidoscopes.
III. Some microscopes are telescopes.
Two 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:
Some tasks are jobs.
No job is a duty.
Conclusions
I. All tasks can never be duties.
II. Some tasks are duties.
Two statements are given followed by conclusions numbered 1 and 2. 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:
1. Some mammals are animals.
2. All animals are birds.
Conclusions:
1. All birds are animals.
2. Some mammals are birds.
Two 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 keys are locks.
All locks are handles.
Conclusions:
I. Some locks are keys.
II. Some handles are keys.
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 switches are fans.
No heater is a fan.
Some wires are heaters.
Conclusions:
I. Some wires are fans.
II. No switch is a heater.
III. No wire is a switch.
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 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 A are S.
II. No D is A.
Conclusions:
I. Some S are A.
II. All S are D.
III. No A is D.
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 land are hard.
II. No stone is land.
Conclusions:
I. Some hard are land.
II. Some stone are hard.
