GATE2016 CE-2: GA-8

Question

Fact 1: Humans are mammals.

Fact 2: Some humans are engineers.

Fact 3: Engineers build houses.

If the above statements are facts, which of the following can be logically inferred?

I. All mammals build houses.

II. Engineers are mammals.

III. Some humans are not engineers.

• A. II only.
• B. III only.
• C. I, II and III.
• D. I only.

Answer 1

We can draw a Venn diagrams for the given facts:

Fact 1 says "Humans are mammals" meaning humans is a subset (may or may not be strict - the diagram shows the not strict version) of mammals.

Fact 2 says "Some humans are engineers" meaning intersection of $H$ and $E$ is non-empty.

Fact 3 says "Engineers build houses" meaning $E$ is a subset (may or may not be strict - the diagram shows the not strict version) of $BH$. 

Now From this diagram we try to get the meaning of given sentences

1. All mammals build houses - False, only if the mammal is a human and he is an engineer he is sure to build a house
2.  Engineers are mammals - False, diagram says some engineers are mammals but does not restrict non mammals to be engineer.
3. Some humans are not engineers - actually this also is not True as we can redraw the diagram making $H$ a subset of $E$ and no Facts are violated. But GATE official key says this is True (Clearly wrong answer key).

No option is correct but official answer key is option B.

Comments

Hey they have clearly mentioned that some human are engineers (means atleast one human is not engineer). So you cant simply put human inside the engineers and declare it as wrong answer. I think Option B is totally correct.

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Thanks sir

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@Satbir – no, I don't think we can’t infer that, “if some humans are engineers then the remaining humans are not engineers”. Because the fact (in the question) only states that, “Some humans are engineers” and doesn’t say anything about the remaining humans. They can or cannot be engineers, but we can’t assume anything for certain. I think it would be wrong.

Answer 2

I think 2nd and 3rd are true but no option matches:(

Comments

not really. I do not think any option here is logically correct.

"Some mammals/humans do not build houses"

If this statement is added, then option B is correct.

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Yes, that is logically not possible. I understand :)

From given 3 facts Fact 1: Humans are mammals. Fact 2: Some humans are engineers. and Fact 3: Engineers build houses , we can draw two Venn Diagrams, one for Mammals another for Build Houses.

see this diagram , 

Mammals consists of Humans - as fact 1 said Humans are mammals means all humans are mammals.

fact 2 Some humans are engineers , see the intersect part, it represents this statement.

fact 3 says Engineers build houses means all engineers build houses .

Now From this diagram we try to get the meaning of given sentences

1.   All mammals build houses.- clearly False , diagram says  some mammals build houses.
2.  Engineers are mammals - false again, diagram says not all engineers are mammals rather some engineers are mammals.
3. Some humans are not engineers - diagram says this is correct . some part of H is not belongs to circle E .

so only statement III is correct , which is option B .

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If we have to have select any answer then option B is okay to be selected otherwise i don't think any statement is logically inferred.

When we say 'Some students are intelligent' we can't claim that some are not.

Answer 3

Fact 1. All Humans are Mammals

           Fact 2.Some Humans are Engineers

 

Fact 3. Engineer build Houses

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1. All mammals build houses is $false$ because only some mammals who are engineers as well as humans build houses.
2. Engineers are mammals is $false$ since only some engineers (who are humans) are mammals.
3. Some humans are not engineers is also $true$ as we can see in fact 2.

$\therefore$ Option B. $Only$ $III$ is the correct answer.

But this answer is not correct.

As mentioned by @Arjun Sir,

In Fact 2 we can say that some human are Engineers but we can't say anything about the remaining humans whether they are engineers or not as it is not mentioned in the question.

So we can't infer that some humans are not engineers as we can have multiple cases for the remaining humans.

Case 1 : The remaining humans are also engineers.

Case 2: The remaining humans are not engineers.

Case 3: Some of the remaining humans are engineers.

Case 4: Some of the remaining humans are not engineers.

Case 5: Some of the remaining humans are engineers and the rest are not engineers.

So, None of the above Statements are correct.

Comments

No, since only some mammals build houses.

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yes, no relation with real world.

Even it is possible "Engineers are not mammal"

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"Engineers are not mammal"

This is false because some humans are engineers and all humans are mammals so

it is possible that some engineers(who are human) are mammals.

"Some Engineers are not mammal" may be correct but we can't assume it because any fact about it is not given.

"Some Engineers are mammal" is correct because we can say that there are some engineers(who are humans ) are mammals because all humans are mammals.