Given the two statements "Some Analysts are fools" and "Some fools are rich", what else can be determined?

Question

an interesting question as below:

The following statements are made:

• Some Analysts are fools.
• Some fools are rich.

Which of the below statements are true?

A. “Some Analysts are rich”

B. “Some rich people are Analysts”

C. “Some Analysts are rich” or “Some rich people are Analysts”

D. Neither “Some Analysts are rich” and “Some rich people are Analysts”

E. Both “Some Analysts are rich” and “Some rich people are Analysts”

This's how i see it: Some Analysts are fools, which are rich. = Some Analysts are rich. Reverse it and read it from the end of sentence to beginning, I got "Some rich people are Analysts". So (E).

Seems E is the best answer. Do you agree? Or, there's no a definite answer?

Answer 1

Note that the formulas "some $X$ are $Y$" and "some $Y$ are $X$" are equivalent: they are both a way of stating that $X \cap Y \neq \emptyset$.

With this in mind, let us use $A,F,R$ for the sets of analysts, fools, and rich people. We know that:

$$A \cap F \neq \emptyset\\ F \cap R \neq \emptyset$$

The question asks which of these is true:

A. $A \cap R \neq \emptyset$
B. $R \cap A \neq \emptyset$
C. $A \cap R \neq \emptyset$ or $R \cap A \neq \emptyset$
D. $A \cap R = \emptyset$ and $R \cap A = \emptyset$
E. $A \cap R \neq \emptyset$ and $R \cap A \neq \emptyset$

Most of these are redundant; we can simplify the question to the following.

A,B,C,E. $A \cap R \neq \emptyset$
D. $A \cap R = \emptyset$

But we can't conclude that either of these is true, since we don't know anything about $A \cap R$.