In the following question below are given some statements followed by some conclusions. 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 cups are plates. II. All plates are red. Conclusions: I. All cups are red. II. All plates are cups.
Only I
Understanding Logic Statements and Conclusions
This question asks us to analyze a set of statements and determine which of the given conclusions logically follows from them, assuming the statements are true, even if they contradict common knowledge. This type of problem falls under the category of logical deduction or syllogism.
Analyzing the Given Statements
We have two statements:
- Statement I: All cups are plates.
- Statement II: All plates are red.
These statements establish a relationship between three categories: Cups, Plates, and Red items.
Statement I tells us that the set of 'Cups' is entirely contained within the set of 'Plates'.
Statement II tells us that the set of 'Plates' is entirely contained within the set of 'Red' items.
Connecting the Statements Logically
If we combine these two statements, we can see a chain of inclusion:
Cups \(\subseteq\) Plates
Plates \(\subseteq\) Red
From these two inclusions, we can logically deduce a relationship between Cups and Red items.
If everything that is a Cup is also a Plate, and everything that is a Plate is also Red, then it must logically follow that everything that is a Cup is also Red. This is a transitive property of inclusion.
Therefore, the logical connection is: Cups \(\subseteq\) Red.
Evaluating the Given Conclusions
Now let's examine each conclusion based on our analysis of the statements:
Conclusion I: All cups are red.
Based on our logical deduction from the statements (Cups \(\subseteq\) Plates and Plates \(\subseteq\) Red implies Cups \(\subseteq\) Red), this conclusion directly follows. If every cup is a plate, and every plate is red, then every cup must necessarily be red.
Evaluation: Logically Follows.
Conclusion II: All plates are cups.
Statement I says "All cups are plates". This means that while all cups are included in the set of plates, the set of plates might be larger than the set of cups. There could be plates that are not cups. For example, if there are 10 cups and they are all plates, there might also be 20 other items that are plates but not cups. Therefore, we cannot conclude that all plates are cups based only on the statements provided.
Evaluation: Does Not Logically Follow.
Conclusion III: All red are plates.
Statement II says "All plates are red". This means that while all plates are included in the set of red items, the set of red items might be larger than the set of plates. There could be red items that are not plates. For example, there might be red apples or red shirts that are not plates. Therefore, we cannot conclude that all red items are plates based only on the statements provided.
Evaluation: Does Not Logically Follow.
Summary of Conclusions
Based on our evaluation:
- Conclusion I: All cups are red. (Logically Follows)
- Conclusion II: All plates are cups. (Does Not Logically Follow)
- Conclusion III: All red are plates. (Does Not Logically Follow)
Only Conclusion I logically follows from the given statements.
Determining the Correct Option
The options provided are:
- Only I
- Only II
- Only III
- All conclusions follow.
Since only Conclusion I logically follows, the correct option is the one stating "Only I".
| Statements | Conclusions | Logically Follows? |
|---|---|---|
| I. All cups are plates. | I. All cups are red. | Yes (via transitivity) |
| II. All plates are red. | II. All plates are cups. | No (Converse of Statement I) |
| III. All red are plates. | No (Converse of Statement II) |
Revision Table: Key Concepts in Syllogism
| Concept | Explanation | Example (from this problem) |
|---|---|---|
| Statement | A premise considered true for logical deduction. | "All cups are plates." |
| Conclusion | A proposition derived from the statements. | "All cups are red." |
| Logically Follows | The conclusion must be true if the statements are true. | Conclusion I follows from Statements I & II. |
| Syllogism | A form of deductive reasoning where a conclusion is drawn from two statements. | The structure of this entire problem. |
| Transitivity | If A implies B, and B implies C, then A implies C. | Cups \(\subseteq\) Plates and Plates \(\subseteq\) Red implies Cups \(\subseteq\) Red. |
| Converse | Reversing the subject and predicate of a statement (e.g., "All A are B" vs "All B are A"). The converse does not necessarily follow. | "All plates are cups" is the converse of "All cups are plates". |
Additional Information: Venn Diagrams for Logic
Venn diagrams are useful visual tools to represent the relationships between sets described in logical statements. For this problem:
- Draw a circle representing 'Red'.
- Inside the 'Red' circle, draw a circle representing 'Plates', because "All plates are red".
- Inside the 'Plates' circle, draw a circle representing 'Cups', because "All cups are plates".
Visually inspecting this diagram would show that the 'Cups' circle is entirely contained within the 'Red' circle, supporting Conclusion I. It would also show that the 'Red' circle and 'Plates' circle are not necessarily the same size (Red could be larger than Plates), and the 'Plates' circle and 'Cups' circle are not necessarily the same size (Plates could be larger than Cups), thus disproving conclusions II and III.
Understanding how to represent 'All X are Y' (X is a subset of Y), 'Some X are Y' (X and Y overlap), and 'No X are Y' (X and Y are separate) using Venn diagrams is crucial for solving syllogism problems.
Similar Questions
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.
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 bottles are glasses.
All glasses are jugs.
Conclusions:
I. All bottles are jugs.
II. All jugs are bottles.
III. Some glasses are bottles.
In the following question below are given some statements followed by some conclusions. 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 pen are black.
II. Some pen are blue.
III. No black is red.
Conclusions:
I. No pen is red.
II. Some blue are black.
III. Some black are pens.In the following question below are given some statements followed by some conclusions. 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 sky are blue.
II. Some blue are soft.
Conclusions:
I. Some sky are soft.
II. Some soft are sky.What will come in the place of the question mark (?) in the following equation if '+' and '-' are interchanged and 'x' and '÷' are interchanged?
108 × 2 − 14 ÷ 6 + 27 = ?
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.
