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:
Some raisins are grapes.
All grapes are apples.
All raisins are fruits.
Conclusions:
I. Some fruits are apples.
II. Some apples are raisins.
Statement Conclusion Logic Problems
This problem involves analyzing logical statements and conclusions to determine their validity. We need to check if the given conclusions can be logically derived from the provided statements.
Analyzing Given Statements
- Statement 1: Some raisins are grapes. (This indicates a partial overlap between the set of raisins and the set of grapes.)
- Statement 2: All grapes are apples. (This means the entire set of grapes is contained within the set of apples.)
- Statement 3: All raisins are fruits. (This means the entire set of raisins is contained within the set of fruits.)
Intermediate Conclusion Derivation
Let's combine the first two statements:
- From Statement 1, we know there exists at least one raisin that is also a grape.
- From Statement 2, we know that every grape is an apple.
- Therefore, if there's a raisin that is a grape, and all grapes are apples, then that specific raisin must also be an apple.
- This allows us to derive an intermediate conclusion: Some raisins are apples.
Conclusion I: Some fruits are apples
Now, let's evaluate Conclusion I using the statements and our derived conclusion:
- We know from Statement 3: All raisins are fruits.
- We derived: Some raisins are apples.
- Since all the raisins mentioned in "All raisins are fruits" are indeed fruits, and we know some of these raisins are also apples, it logically follows that those specific raisins represent fruits that are also apples.
- Therefore, Some fruits are apples. Conclusion I logically follows.
Conclusion II: Some apples are raisins
Let's evaluate Conclusion II:
- We derived the statement: Some raisins are apples.
- This statement means that there is at least one member that belongs to both the 'raisins' category and the 'apples' category.
- The statement "Some A are B" is logically equivalent to "Some B are A".
- Thus, "Some raisins are apples" logically implies Some apples are raisins. Conclusion II logically follows.
Overall Validity
Based on the step-by-step analysis:
- Conclusion I is valid.
- Conclusion II is valid.
Both conclusions logically follow from the given statements.
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:
All dancers are talented.
Some girls are dancers.
Conclusions:
I. Some girls are talented.
II. All talented are girls.
III. All girls are talented.
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 directors are actors.
No actor is a producer.
All choreographers are directors.
Conclusions:
I. No choreographer is producer.
II. Some actors are choreographers.
III. No director is a producer.
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.
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 employees are tax-payers.
Some employees are farmers.
Some farmers are doctors.
Conclusions:
I. No farmer is a tax-payer.
II. Some farmers are tax-payers.
