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 police officer is a doctor. Some doctors are specialists. All engineers are doctors. Conclusions: I. Some engineers are police officers. II. No engineer is a police officer. III. Some doctors are engineers.
Understanding Logical Statements and Conclusions
This problem asks us to analyze a set of statements and determine which of the given conclusions logically follow from them. We must assume the statements are true, even if they contradict common knowledge. This type of problem is based on deductive reasoning.
Analyzing the Statements
Let's break down the information provided in each statement:
- Statement 1: No police officer is a doctor. This establishes a complete separation between the set of police officers and the set of doctors. There is no overlap between these two groups.
- Statement 2: Some doctors are specialists. This means there is an overlap between the set of doctors and the set of specialists. It implies that some individuals are both doctors and specialists. However, it doesn't say anything about doctors who are *not* specialists, or specialists who are *not* doctors.
- Statement 3: All engineers are doctors. This is a crucial statement. It tells us that the entire set of engineers is contained within the set of doctors. Every single engineer is also a doctor. This means engineers form a subset of doctors.
Representing the Relationships
We can visualize these relationships using sets:
- Police Officers and Doctors are disjoint sets.
- Doctors and Specialists have an intersection (some members are common).
- Engineers are entirely inside the set of Doctors.
Combining these, since Engineers are inside Doctors, and Police Officers are outside Doctors, Engineers must also be outside Police Officers.
| Relationship | Description |
|---|---|
| Police Officer <--> Doctor | No Overlap (Mutually Exclusive) |
| Doctor <--> Specialist | Partial Overlap (Some are common) |
| Engineer <--> Doctor | Engineer is a Subset of Doctor (All Engineers are Doctors) |
Evaluating the Conclusions
Now let's examine each conclusion based on our analysis of the statements:
- Conclusion I: Some engineers are police officers.
From Statement 3, we know that all engineers are doctors. From Statement 1, we know that no police officer is a doctor. If engineers are doctors, and no doctors are police officers, then no engineers can be police officers. Therefore, the conclusion "Some engineers are police officers" is false.
- Conclusion II: No engineer is a police officer.
As reasoned above, because all engineers are doctors (Statement 3), and no doctor is a police officer (Statement 1), it logically follows that no engineer can be a police officer. This conclusion is true.
- Conclusion III: Some doctors are engineers.
Statement 3 says that "All engineers are doctors". If the entire group of engineers is part of the group of doctors, then it must be true that some members of the group of doctors are engineers (specifically, all the engineers). This is a direct implication of "All A are B" → "Some B are A". Therefore, the conclusion "Some doctors are engineers" is true.
Determining Which Conclusions Follow
Based on our evaluation:
- Conclusion I: Does not follow.
- Conclusion II: Follows.
- Conclusion III: Follows.
Thus, only conclusions II and III logically follow from the given statements.
Revision Table: Logical Reasoning Summary
| Statement/Conclusion | Relationship/Truth | Analysis |
|---|---|---|
| S1: No PO is a Doctor | PO ∩ Doctor = ∅ | Disjoint sets |
| S2: Some Doctors are Specialists | Doctor ∩ Specialist ≠ ∅ | Partial overlap |
| S3: All Engineers are Doctors | Engineer ⊂ Doctor | Subset relationship |
| C1: Some Engineers are PO | False | Engineers are Doctors, Doctors are not PO. So Engineers are not PO. |
| C2: No Engineer is a PO | True | Directly follows from Engineers ⊂ Doctors and PO ∩ Doctor = ∅. |
| C3: Some Doctors are Engineers | True | Directly follows from Engineer ⊂ Doctor (If All A are B, then Some B are A). |
Additional Information: Syllogisms and Deductive Reasoning
This question is an example of a syllogism, a type of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are assumed to be true. The statements provide the premises, and we test whether the conclusions are valid deductions.
Key concepts in such problems include:
- Universal Negative (No A is B): States that two categories have no members in common. Implies "No B is A".
- Particular Affirmative (Some A are B): States that there is at least one member common to both categories. Implies "Some B are A".
- Universal Affirmative (All A are B): States that every member of category A is also a member of category B. Implies "A is a subset of B". This also implies "Some A are B" and "Some B are A".
- Deductive Reasoning: Moving from general principles (statements) to specific conclusions. If the premises are true, the conclusion must be true if the logic is valid.
Using Venn diagrams or just carefully tracking the relationships between the categories helps in solving these problems accurately.
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 follows from the statements.
Statements:
All lemons are plums.
All plums are dates.
Some dates are mangoes.
Conclusions:
I. Some lemons are mangoes.
II. Some mangoes are plums.
III. All lemons are dates.
IV. Some mangoes are dates.
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 cards are postcards.
Some cards are envelopes.
All envelopes are copies.
Conclusions:
I. Some copies are envelopes.
II. Some postcards are copies.
III. Some cards are copies.
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 spices are medicines.
All medicines are chemicals.
No chemical is cheap.
Conclusions:
I. Some medicines are cheap.
II. Some chemicals are spices.
III. No medicine is cheap.
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 Z are X.
II. Some W are Z.
Conclusions:
I. Some X are not W.
II. Some X are not Z.
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 mobiles are flat.
All flats are TVs.
Some TVs are plastic.
Conclusions:
I. Some mobiles are TVs.
II. Some flats are plastic.
III. All plastic are TVs.
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 cup is a plate.
All cups are utensils.
Some bottles are cups.
Conclusions:
I. Some utensils are not plates.
II. Some bottles are not plates.
III. No bottle is a plate.
Three statements are followed by three conclusions numbered I, II and III. You have to consider these 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 given statements.
Statements:
Some watches are projectors.
All projectors are lamps.
Some heaters are watches.
Conclusions:
(I) Some heaters are projectors.
(II) Some lamps are heaters.
(III) All watches are lamps
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. No A is B.
II. No C is B.
Conclusion:
I. Some C are not A.
II. Some A are B.
Consider the given statement(s) to be true and decide which of the given conclusions/assumptions can definitely be drawn from the given statement.
Statement 1: All elephants are men.
Statement 2: All men are socks.
Conclusion I: Some socks are elephants.
Conclusion II: All elephants are socks.In the question, a word is represented by only one set of numbers as given in any one of the alternatives. The sets of numbers given in the question are represented by two classes of alphabets in two matrices given below. The columns and rows of Matrix 1 are numbered from 0 to 4 and that of Matrix II are numbered from 5 to 9. A letter from these matrices can be represented first by its row and next by its column, e.g., 'A' can be represented by 04, 12, etc., and 'U' can be represented by 59, 66, etc. Similarly, you have to identify the set for the word 'ROOT'.
MATRIX I
0 1 2 3 4 0 B C D E A 1 D E A B C 2 C D E A B 3 A B C D E 4 E A B C D MATRIX II
5 6 7 8 9 5 R O N T U 6 T U R O N 7 O N T U R 8 U R O N T 9 N T U R O
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.
