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 conclusion(s) logically follow(s) from the statements.
Statements:
All cats are bills.
No snakes are bills.
All snakes are dolls.
Conclusions:
(I) Some dolls are not cats.
(II) No bills are dolls.
Only conclusion (I) follows.
This problem involves logical reasoning based on given statements and conclusions. We need to determine which conclusion(s) logically follow from the statements, assuming the statements are true.
Analyzing the Statements
Let's break down each statement and understand the relationship between the sets mentioned (Cats, Bills, Snakes, Dolls).
- Statement 1: All cats are bills.
This means the set of 'cats' is entirely contained within the set of 'bills'. If something is a cat, it must also be a bill.
We can represent this as: Cats ⊆ Bills.
- Statement 2: No snakes are bills.
This means the set of 'snakes' and the set of 'bills' are completely separate; they have no members in common. If something is a snake, it cannot be a bill, and vice versa.
We can represent this as: Snakes ∩ Bills = ∅ (empty set).
- Statement 3: All snakes are dolls.
This means the set of 'snakes' is entirely contained within the set of 'dolls'. If something is a snake, it must also be a doll.
We can represent this as: Snakes ⊆ Dolls.
Evaluating the Conclusions
Now, let's examine each conclusion based on the relationships established by the statements.
Conclusion (I): Some dolls are not cats.
Let's combine the information:
- From Statement 3, we know that all snakes are dolls (Snakes ⊆ Dolls).
- From Statement 2, we know that no snakes are bills (Snakes ∩ Bills = ∅).
- From Statement 1, we know that all cats are bills (Cats ⊆ Bills).
Since all snakes are dolls (Statement 3) and no snakes are bills (Statement 2), we can infer that the snakes, which are a type of doll, are definitely not bills.
Furthermore, since all cats are bills (Statement 1), and we just established that snakes (which are dolls) are not bills, it follows that these snakes cannot be cats.
Therefore, there exists at least one group within the set of 'dolls' (namely, the snakes) that is not part of the set of 'cats'. This directly supports the conclusion that 'Some dolls are not cats'.
Conclusion (I) logically follows.
Conclusion (II): No bills are dolls.
Let's consider the relationships again:
- Cats are a subset of Bills (Cats ⊆ Bills).
- Snakes are a subset of Dolls (Snakes ⊆ Dolls).
- Snakes and Bills are disjoint (Snakes ∩ Bills = ∅).
These statements tell us that the snakes (which are dolls) are separate from bills. However, they do not provide information about whether other members of the 'Dolls' set (those that are not snakes) might also be members of the 'Bills' set.
It is possible for an object to be both a bill and a doll, as long as it is not a snake. For example, imagine a doll that is also a bill, but this doll is neither a cat nor a snake. This scenario does not contradict any of the given statements.
Because we can conceive of a situation where a bill is also a doll (without violating the given statements), the conclusion 'No bills are dolls' cannot be definitively proven true.
Conclusion (II) does not necessarily follow.
Final Determination
Based on the analysis, only Conclusion (I) logically follows from the given statements.
Important Questions from Statement and Conclusions
- Select the conclusion, from among the following options, which logically follows from the given statement.
Statement:
One must work hard to be successful in life. - Read the given statement 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.
Statement:
Doing exercises in the morning is good for health.
Conclusion 1:
Health can be maintained by doing exercises only.
Conclusion 2:
Without doing exercises, people will become ill. - Read the given statements and conclusions carefully and select the conclusion(s) that logically follow(s) from the statements.
Statements:
Indian Education is lacking quality due to low funds. India is allocating additional funds to education sector.
Conclusions:
I. There would be improvement in quality of education in India.
II. Only funds can improve the education standards. - In the question below is given a statement followed by two conclusions numbered I and II. You have to assume everything in the statement to be true, then consider the two conclusions together and decide which of them logically follows beyond a reasonable doubt from the information given in the statement.
Statement:
If A was there in XYZ cricket team, XYZ would have been the invincible team in the world.
Conclusion:
I. Without A, XYZ cannot win.
II. A is more talented than anyone else in the XYZ cricket team - Read the given statement carefully and select from among the given options, the conclusion that logically follows from the statement.
Statement:
Ability is a poor man's wealth.
