# Which can be logically inferred from the given statements?

1. All women are entrepreneurs.
2. Some women are doctors.

Which of the following conclusions can be logically inferred from the above statements?
(A) All women are doctors.
(B) All doctors are entrepreneurs.
(C) All entrepreneurs are women.
(D) Some entrepreneurs are doctors.

-As far as I could think, (A) is clearly not the option.
-If I choose (B), that would also mean doctors who are not women are also entrepreneurs.
-If I choose (C), that would mean men cannot be entrepreneurs.
-And if I choose (D), that is most likely the answer, but I'm still not sure.

The doubt is whether I can consider men/non-women while solving such questions.

(D) is the correct answer as:-

(A). All women might not be doctors as it is not explicitly mentioned.
(B). There might be doctors which are not women (men, transgender?). Since nothing is mentioned explicitly about the general populace of doctors, we cannot assume that there are only women doctors.
(C). There might be entrepreneurs which are not women.
(D). This is correct as all women are entrepreneurs. Also, some of these women entrepreneurs are doctors.

• It should be noted that D is only correct if the set of women is not empty. Probably a safe assumption, but it is a hidden assumption. Commented Jan 22, 2015 at 5:04

This is a syllogism of type Bocardo. Check the Venn diagram on the Wikipedia page

Let's make two cases.

Case 1: Men/non-women must be considered. Then you have correctly found reasons why A,B and C cannot be true. Hence D is your only option, and is correct in my opinion as well.

Case 2: Only women are considered. In this case, D is no less correct. However, C and B also becomes true since we are only considering the set of women, and all women are entrepreneurs. A is still clearly not true.

In case $2$ you end up with three correct deductions, while in case $1$ you end up with $1$ correct deduction. Given that this is a multiple choice question and multiple choice questions almost always have exactly one answer, I'd bet that case $1$ is what you want to consider.

• Since from the given statements you cannot infer that no objects exist that are doctor bot not women etc., case 2 is not relevant to the question. Commented Jan 20, 2015 at 16:03
• @HagenvonEitzen that is a good point, and it succinctly demonstrates the irrelevancy of case 2. Commented Jan 20, 2015 at 16:06

I believe D is the correct answer because all of the female doctors are entrepreneurs and some females are not doctors.

In fact, in given statement #2, you can replace "women" with "entrepreneurs" and arrive exactly at answer D.

D - Some entrepreneurs are doctors - is the correct answer:

Let's call the set of Women W, the set of Doctors D and the set of Entrepreneurs E.

• you need to demonstrate: ED ⊊ ∅.

By 1. you have: WE, while by 2. you have: ∅ ⊋ WD.

So ∅ ⊋ WDWDED.

And by the last chain you have your thesis: ∅ ⊋ ED.

• Check the direction of the inclusions: there is no strict subset of the empty set. Also, the step $W \cap D \supseteq W \cap D$ looks pointless.
– chi
Commented Jan 20, 2015 at 23:02