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 Logical Reasoning Statements and Conclusions

This problem involves analyzing logical relationships between different categories based on given statements. We need to determine which of the provided conclusions can be definitively inferred as true, assuming the initial statements are correct.

Analyzing the Given Statements

The core task is to interpret the premises and derive logical consequences. Let's break down the statements:

Statement 1: All cucumbers are onions.
Statement 2: All cucumbers are mangoes.

These statements establish a relationship where the set of 'cucumbers' is entirely contained within the set of 'onions' and also entirely contained within the set of 'mangoes'. This implies that anything identified as a 'cucumber' must necessarily be both an 'onion' and a 'mango'.

Evaluating Conclusion (I): Some onions are mangoes

Detailed Analysis:

From Statement 1, we know that every cucumber is an onion.
From Statement 2, we know that every cucumber is a mango.
If we consider the category 'cucumbers', all individuals within this category possess both properties: being an onion and being a mango.
Therefore, there must exist at least some individuals (namely, the cucumbers) that are common to both the 'onions' category and the 'mangoes' category.
This directly confirms that "Some onions are mangoes".

Result: Conclusion (I) logically follows from the given statements.

Evaluating Conclusion (II): All mangoes are cucumbers

Detailed Analysis:

The statements tell us that 'cucumbers' are a subset of 'mangoes' (All C are M).
However, they do not provide any information suggesting the reverse relationship, i.e., that 'mangoes' are a subset of 'cucumbers' (All M are C).
It is entirely possible for the set of 'mangoes' to be larger than the set of 'cucumbers'. There could be mangoes that are not cucumbers, while still satisfying the condition that all cucumbers are mangoes.
For instance, consider a scenario where there are 10 cucumbers, 20 onions, and 30 mangoes. If these 10 cucumbers are indeed onions and also mangoes, the original statements hold true. Yet, the remaining 20 mangoes are not cucumbers.
Thus, we cannot logically conclude that "All mangoes are cucumbers" based solely on the given statements.

Result: Conclusion (II) does not logically follow from the given statements.

Summary of Logical Deduction

Based on the step-by-step analysis, only the first conclusion is a valid deduction from the provided statements.

Statement	Logical Representation
All cucumbers are onions.	If $x$ is a cucumber, then $x$ is an onion. ($C \subseteq O$)
All cucumbers are mangoes.	If $x$ is a cucumber, then $x$ is a mango. ($C \subseteq M$)

Conclusion	Validity	Explanation
(I) Some onions are mangoes.	Follows	Since all cucumbers are both onions and mangoes, the intersection of onions and mangoes must contain cucumbers. Therefore, at least some onions are mangoes. ($C \subseteq O$ and $C \subseteq M \implies C \subseteq (O \cap M)$)
(II) All mangoes are cucumbers.	Does Not Follow	The statements only establish that cucumbers are a subset of mangoes ($C \subseteq M$). They do not imply that mangoes are a subset of cucumbers ($M \subseteq C$). The set of mangoes could be larger than the set of cucumbers.