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: Some tents are cones. All the cones are motels. Conclusions: All the motels are cones.
Only conclusion (II) follows.
Analyzing Logical Statements and Conclusions
This problem requires us to analyze given statements and determine which of the provided conclusions logically and definitely follow from these statements. This type of question is common in logical reasoning sections of exams.
Understanding the Statements
We are given two statements:
- Some tents are cones.
- All the cones are motels.
We must assume these statements are true, even if they contradict common knowledge.
Evaluating the Conclusions
We have two conclusions to evaluate:
- All the motels are cones.
- Some tents are motels.
Evaluation of Conclusion I: All the motels are cones.
Let's consider the relationship between cones and motels based on Statement 2: "All the cones are motels." This means that the set of cones is entirely contained within the set of motels. We can represent this relationship as \(Cones \subseteq Motels\).
Conclusion I states the reverse: "All the motels are cones," which means \(Motels \subseteq Cones\).
Based on Statement 2, it is possible for there to be motels that are not cones. For example, if we have Cones = {A, B} and Motels = {A, B, C}, Statement 2 ("All the cones are motels") is true because A and B are in the set of Motels. However, Conclusion I ("All the motels are cones") is false because C is a motel but not a cone.
Therefore, Conclusion I does not logically or definitely follow from the given statements.
Evaluation of Conclusion II: Some tents are motels.
Let's combine the information from both statements:
- Some tents are cones. (\(Tents \cap Cones \neq \emptyset\))
- All the cones are motels. (\(Cones \subseteq Motels\))
Statement 1 tells us there is at least one element that is both a tent and a cone. Let's call this common element (or set of elements) X. So, X is a tent AND X is a cone.
Statement 2 tells us that anything that is a cone is also a motel. Since X is a cone (from Statement 1), it must also be a motel (from Statement 2).
So, X is a tent AND X is a cone AND X is a motel. Since we know X exists (from Statement 1, "Some tents are cones"), there is at least one element that is both a tent and a motel.
This means that the intersection of Tents and Motels is not empty (\(Tents \cap Motels \neq \emptyset\)). This is exactly what Conclusion II states: "Some tents are motels."
Therefore, Conclusion II logically and definitely follows from the given statements.
Summary of Conclusions
Based on our analysis:
- Conclusion I ("All the motels are cones") does not follow.
- Conclusion II ("Some tents are motels") follows.
Thus, only Conclusion (II) follows from the given statements.
Revision Table: Syllogism Analysis
| Statement Type | Representation | Example |
|---|---|---|
| All A are B | A \(\subseteq\) B | All cats are animals. |
| Some A are B | A \(\cap\) B \(\neq \emptyset\) | Some students are tall. |
| No A are B | A \(\cap\) B = \(\emptyset\) | No dogs are birds. |
| Some A are not B | Subset of A is outside B | Some food is not healthy. |
Additional Information: Principles of Logical Deduction
Syllogisms involve deductive reasoning, where if the premises (statements) are true, the conclusion must also be true. A common pattern used here is similar to transitivity:
- If "Some A are B" and "All B are C", then it logically follows that "Some A are C".
In this problem:
- A = Tents
- B = Cones
- C = Motels
The statements are "Some Tents are Cones" and "All Cones are Motels". Applying the principle, we can deduce "Some Tents are Motels", which is Conclusion II.
It's important to distinguish between "All A are B" and "All B are A". These are usually not equivalent. Just because all dogs are animals does not mean all animals are dogs.
Similar Questions
Read the given statements and conclusions carefully. Assuming that the information given in the statements are 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 colours are reds.
2. Some reds are greens.
3. All greens are dark.
Conclusions:
I. All colours being dark is a possibility
II. Some reds are dark.
III. Some reds are colours.
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 chairs are cupboards.
Some chairs are radios.
No chair is piano.
Conclusions:
I. Some radios are cupboards.
II. Some cupboards are not piano.
III. Some radios are not piano.
Read the given statements and conclusions carefully. Assuming that the information given in the statements is ture, 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 aero-planes are cars.
All helicopters are cars.
All cars are scooters.
Conclusions:
1. All helicopters are scooters.
2. Some aero-planes are helicopters.
3. No aero-plane is helicopter.
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 books are grapes.
Some grapes are fruits.
Conclusions:
I. No grape is a book.
II. Some grapes are books.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 fans are radio.
All fans are light.
Conclusion I: Some radios are light.
Conclusion II: Some lights are radio.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 scissors are spoons.
All spoons are plates.
Conclusions:
1. Some plates are spoons.
2. Some spoons are scissors.
3. Some plates are scissors.
Read the given statements and conclusions carefully. Assuming that the information given in the statement 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 boat is ship.
Some ships are rods.
Conclusions:
I. No boat is rod.
II. Some rods are ships.
III. Some rods are not boats.
The statements below are followed by four conclusions labelled I, II, III and IV. 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:
No laptop is notebook.
All notebooks are tablets.
Conclusions:
I. No laptop is tablets.
II. No tablets is laptop.
III. Some tablets are notebooks.
IV. All the tablets are notebooks.
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 cats are hens.
All hens are goat.
Conclusions:
Conclusion I: Some hens are cat.
Conclusion II: Some goats are hens.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 pictures are prizes.
Some prizes are erasers.
Some erasers are helmets.
Conclusions:
I. Some helmets are Prizes.
II. Some erasers are pictures.
III. No helmet is a Prize.
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.
