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
- Statement: Resilience supports recovery from setbacks, but without learning from mistakes, progress may stall.
Conclusion I: Resilience alone may not ensure progress.
Conclusion II: Learning from mistakes guarantees recovery.
Which conclusion follows? - Read the following statement carefully and identify the conclusion that follows.
Statement: "Government should provide free education to all children up to the age of 14."
Conclusions:
I. Education is a basic right of every child.
II. A large number of children are deprived of basic education. - Statement: Ethical behavior fosters trust, but profitability often relies on operational efficiency as well.
Conclusion I: Ethical behavior is not sufficient for profitability.
Conclusion II: Operational efficiency is required for trust.
Which conclusion follows? - Read the following statement carefully and identify the conclusion that follows.
Statement: "Online learning has increased student participation."
Conclusions:
I. Students did not participate in physical classes.
II. Technology helps in engaging students. - S1: Failure is the stepping stone to success.
S2: Successive failures are not however successive stepping stones to success.
The second sentence:
