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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.

  1. II only.
  2. III only.
  3. I, II and III.
  4. I only.
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3 Answers

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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.

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I think 2nd and 3rd are true but no option matches:(
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Fact 1. All Humans are Mammals
           Fact 2.Some Humans are Engineers

 

Fact 3. Engineer build Houses


  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.

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