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 calculators are lanterns. Some lanterns are mobiles. Conclusions: I. All mobiles are calculators. II. No mobile is a lantern. III. Some mobiles are lanterns.
Only conclusion III follows
This question asks us to analyze given statements and determine which of the provided conclusions logically follow. This is a common type of logic puzzle called a syllogism, often involving relationships between categories or groups.
Analyzing the Statements
We are given two statements:
- Some calculators are lanterns.
- Some lanterns are mobiles.
In logic problems like this, we must accept the statements as true, even if they seem unusual in the real world.
Let's break down what these statements mean:
- Statement 1: There is at least one item that is both a calculator and a lantern. There might be more, but definitely at least one. We can represent this as Calculators $\cap$ Lanterns $\neq$ $\emptyset$.
- Statement 2: There is at least one item that is both a lantern and a mobile. Again, there might be more, but at least one exists. We can represent this as Lanterns $\cap$ Mobiles $\neq$ $\emptyset$.
It's important to note that these statements only give us "some" relationships. They don't tell us about "all" items in a category or rule out connections between categories not explicitly mentioned.
Evaluating the Conclusions
Now let's examine each conclusion based on the truth of the statements.
Conclusion I: All mobiles are calculators.
This conclusion claims a universal relationship: every single mobile is also a calculator. The statements tell us about an overlap between calculators and lanterns, and an overlap between lanterns and mobiles. However, they provide no direct information linking *all* mobiles to *all* calculators, or even any specific mobile to a calculator, except indirectly through lanterns. It is possible to draw diagrams or visualize scenarios where Statement 1 and Statement 2 are true, but not all mobiles are calculators. For example, there could be some calculators that are lanterns, and some lanterns that are mobiles, but the group of items that are mobiles might not overlap with the group of items that are calculators at all (except possibly via the lantern link, but even that isn't guaranteed for *all* mobiles). Therefore, this conclusion does not logically follow from the statements.
Conclusion II: No mobile is a lantern.
This conclusion claims there is no overlap between mobiles and lanterns. This directly contradicts Statement 2, which explicitly says "Some lanterns are mobiles." The statement "Some A are B" logically implies that "Some B are A". So, "Some lanterns are mobiles" is equivalent to "Some mobiles are lanterns." Therefore, Conclusion II, stating "No mobile is a lantern," is false if Statement 2 is true. Since we assume the statements are true, this conclusion does not logically follow; in fact, it is contradicted by the statements.
Conclusion III: Some mobiles are lanterns.
This conclusion states that there is at least one item that is both a mobile and a lantern. As discussed when analyzing Conclusion II, Statement 2 is "Some lanterns are mobiles." The logical implication of "Some lanterns are mobiles" is that "Some mobiles are lanterns." This conclusion is a direct restatement (in a logically equivalent form) of information given in Statement 2. Therefore, this conclusion logically follows from the statements.
Determining the Correct Answer
Based on our analysis:
- Conclusion I: Does not follow.
- Conclusion II: Does not follow (contradicted).
- Conclusion III: Logically follows.
Only Conclusion III follows from the given statements.
| Statement | Meaning | Equivalent Form |
|---|---|---|
| Some calculators are lanterns. | Calculators and Lanterns sets overlap. | Some lanterns are calculators. |
| Some lanterns are mobiles. | Lanterns and Mobiles sets overlap. | Some mobiles are lanterns. |
| Conclusion | Analysis | Logically Follows? |
|---|---|---|
| I. All mobiles are calculators. | Statements give "some" relations, no direct link between All Mobiles and Calculators. | No |
| II. No mobile is a lantern. | Contradicts Statement 2 ("Some lanterns are mobiles" $\equiv$ "Some mobiles are lanterns"). | No |
| III. Some mobiles are lanterns. | Equivalent to Statement 2. | Yes |
Thus, only Conclusion III is logically valid.
Revision Table: Statements & Conclusions Logic
| Concept | Description | Example |
|---|---|---|
| Syllogism | A type of logical argument where a conclusion is derived from two or more statements. | Statements: All A are B. All B are C. Conclusion: All A are C. |
| "Some" Statements | Indicates an overlap between two categories, meaning at least one element belongs to both. Does not provide information about the entire set. | "Some cats are black." Means there's at least one black cat. Doesn't say anything about all cats or all black things. |
| "All" Statements | Indicates that every member of one category is also a member of another. | "All dogs are mammals." Means every dog belongs to the set of mammals. |
| "No" Statements | Indicates no overlap between two categories. | "No birds are fish." Means the set of birds and the set of fish are entirely separate. |
Additional Information: Venn Diagrams in Logic
Venn diagrams are helpful tools for visualizing syllogisms. Each statement translates into how circles representing categories overlap. For our statements:
- "Some calculators are lanterns" means the Calculator circle and the Lantern circle overlap, and the overlapping area is not empty.
- "Some lanterns are mobiles" means the Lantern circle and the Mobile circle overlap, and the overlapping area is not empty.
When you draw these, you'll see the Calculator and Mobile circles might or might not overlap. Since we can draw a scenario where they don't (while still satisfying the statements), "All mobiles are calculators" and "No mobile is a calculator" cannot be concluded. The overlap between Lanterns and Mobiles, however, is required by Statement 2, directly supporting Conclusion III.
Remember, a conclusion follows only if it is true in *every* possible scenario where the statements are true.
Similar Questions
- In this question, 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 follows/follow from the statements.
Statements:
I. Some tables are chairs.
II. All desks are chairs.
III. No chair is a fan.
Conclusions:
I. Some desks are tables.
II. Some fans are desks. - In this question, 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 follows/follow from the statements.
Statements:
I. All tables are charts.
II. Some charts are graphs.
III. Some graphs are pies.
Conclusions:
I. No table is a graph.
II. Some pies are charts. - In this question, 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 follows/follow from the statements.
Statements:
All boxes are squares.
All squares are rectangles.
All rectangles are figures.
Conclusions:
I. All boxes are figures.
II. All squares are figures. 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 given conclusions logically follow(s) from the statements.
Statements:
Some fractions are marts.
No mart is a stud.
All studs are frogs.
Conclusions:
I. Some fractions are definitely not studs.
II. Some fractions are definitely not marts.
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 parrots are crows.
All crows are eagles.
Conclusions:
I. All parrots are eagles.
Il. Some crows are parrots.
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 from the statements.
Statements:
No certificate is a paper.
All pens are papers.
Conclusions:
I. No certificate is a pen.
II. No paper is a certificate.
III. Some papers are pens.
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 officers are clerks.
All clerks are workers.
Conclusions:
I. Some workers are officers.
Il. No worker is an officer.
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 chargers are phones.
No phone is a watch.
Conclusions:
I. All phones are chargers.
Il. No charger is a watch.
Ill. No watch is a phone.
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 wallets are bags.
Some bags are envelopes.
Conclusions:
I. Some wallets are envelopes.
II. No wallet is an envelope.
III. No bag is an envelope.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:
Some boys are brave.
All boys are punctual.
Conclusions:
I. Some brave are boys.
II. Some punctual are boys.
III. Some brave are punctual.
Important Questions from 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:
Some students are players.
All players are male.
Conclusions:
I. All males are players.
II. Some males are students.
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 creative is an employer.
All experts are creative.
All workers are experts.
Conclusions:
I. No employer is an expert.
II. No worker is an employer.
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 actors are choreographers.
All choreographers are producers.
Not a single producer is a director.
Conclusions:
I. Some actors are directors.
II. Not a single actor is a director.
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 flowers are tulips.
2. No tulips are whites.
Conclusions :
I. All tulips are flowers.
II. Some tulips are whites.
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 rats are dogs.
Some rats are hens.
Conclusions:
I. Some rats are dogs.
II. Some hens are rats.
