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 canals are rivers.
Some rivers are oceans.
Some oceans are ponds.

Conclusions:
(I) All ponds are rivers.
(II) All oceans are canals.

Question

Prepp Team · Accepted Answer

Understanding the Syllogism Statements

This problem involves logical reasoning, specifically analyzing syllogisms. We are given two statements and two conclusions. We need to determine which conclusion(s) must be true if the statements are true.

Statements provided:

Statement 1: All hens are drivers.
Statement 2: All fruits are drivers.

Let's represent these statements using sets:

Let H be the set of all hens.
Let D be the set of all drivers.
Let F be the set of all fruits.

Based on the statements:

Statement 1 means H is a subset of D (H ⊆ D). Every member of set H is also a member of set D.
Statement 2 means F is a subset of D (F ⊆ D). Every member of set F is also a member of set D.

Visually, we can imagine a large circle representing 'drivers'. Inside this large circle, there are two smaller circles, one for 'hens' and one for 'fruits'. Both smaller circles are completely within the 'drivers' circle.

Analyzing the Conclusions

Now let's evaluate each conclusion based on the information from the statements.

Conclusion (I): Some hens are fruits.

This conclusion claims that there is an overlap between the set of hens (H) and the set of fruits (F). In set notation, it means H ∩ F ≠ ∅.

From the statements, we know that both hens (H) and fruits (F) are separate groups within the larger group of drivers (D). The statements do not provide any information about whether the 'hens' group overlaps with the 'fruits' group. It's possible they are completely separate groups within the 'drivers' category. Therefore, this conclusion does not necessarily follow logically from the given statements.

Conclusion (II): Some drivers are fruits.

This conclusion claims that there are members of the set 'drivers' (D) who are also members of the set 'fruits' (F). In set notation, it means D ∩ F ≠ ∅.

We know from Statement 2 that "All fruits are drivers" (F ⊆ D). If every single fruit is also a driver, it logically follows that at least some drivers must be fruits. Specifically, all the members of the set F are drivers. Therefore, this conclusion logically follows from the statements.