TIFR CSE 2018 | Part A | Question: 11

Question

We are given a (possibly empty) set of objects. Each object in the set is colored either black or white, is shaped either circular or rectangular, and has a profile that is either fat or thin, Those properties obey the following principles:

1. Each white object is also circular.
2. Not all thin objects are black.
3. Each rectangular object is also either thin or white or both thin and white.

Consider the following statements:

1. If there is a thin object in the set, then there is also a white object.
2. If there is a rectangular object in the set, then there are at least two objects.
3. Every fat object in the set is circular.

Which of the above statements must be TRUE for the set?

• A. $(i)$ only
• B. $(i) \text{ and } (ii)$ only
• C. $(i) \text{ and } (iii)$ only 
• D. None of the statements must be TRUE
• E. All of the statements must be TRUE

Comments

option B

Answer 1

1. Each white object is also circular.
2. Not all thin objects are black.
3. Each rectangular object is also either thin or white or both thin and white.

Consider the following statements:

1. If there is a thin object in the set, then there is also a white object.
   "Not all thin objects are black." means there is a thin object which is not black and white is the only other color possible. So, this is TRUE.
2. If there is a rectangular object in the set, then there are at least two objects.
   "Each rectangular object is also either thin or white or both thin and white". So each rectangular object is either thin or white or both. But "Each white object is also circular" means, a rectangular object cannot be white and hence it must be thin. Now, "Not all thin objects are black" means there is a white object and this is circular as per $(1).$ So, if there is a rectangular object there are at least two objects and this statement is TRUE.
3. Every fat object in the set is circular.
   Can a fat object be rectangular? No, because as seen in $(ii)$ a rectangular object must be thin (as it cannot be white). This means every fat object must be circular as there are no other possibilities. This statement is also TRUE.

Correct Answer: E.

Comments

"Not all thin objects are black."

isnt it implies that some thin objects are black and some thin objects are white ,

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This means every fat object must be circular  

How? Are these object properties exclusive? Means if one is black then the other objects can’t be black?

 

 

 

Answer 2

Statement iii says every fat object is circular. You cannot counter it by saying "thin object can also be circular". Yes thin can be circular. But statement does not say every circular is fat. It says every fat is circular.

Now, every rectangular is thin. Which means there is no fat object which is rectangular. Thus if any fat object is present then it must be circular.

Ans E: All the options are correct

Comments

Not necessary, read first line carefully

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But what is the problem is fat object is not present ?

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1. Each white object is also circular
2. Each rectangular object is also either thin or white or both thin and white.

these two are contradictory statement

So, I too think  either C) will be correct or E) will be correct

Because I can take a set like 3 elements

${\color{Blue} {thin, black}}\rightarrow {\color{Blue} {circular}}$

${\color{red} {thin, white}}\rightarrow {\color{red} {rectangular}}$

${\color{purple} {fat, white}}\rightarrow {\color{purple} {circular}}$

no more item needed

Answer 3

Option 1 is true

This can be verify by statement 2 not all this object are black

Which means there is at least one object which is thin and white

 

Option 2 is true

If there is an rectangle object then it must be thin and black in color and also there must be another object which has to be white and circular as per option one

( Rectangle object cannot be white as it is contractions )

Or there will be no rectangle object at all which also makes statement 2 true
 

Option B is true

Comments

How can you say option 2 is wrong because of statement 3 ?

If there is a rectangular object then if

It is thin then atleast one thin is white. And white are circular thus 2 objects are present i.e rectangular and circular

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Nice thinking but what if it's thin and white both?

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It says either thin OR white OR Both.

But we can never have the last two condition because every white is circular.

So we can conclude that every rectangular is black.

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I think Option B, is correct

Statement (i) says if there is thin object in the set then there is also a white object

Since its given that not all thin objects are black, then there exists at least one white object in the set if it contains a thin object, making statement one true

Statement (ii) says if there is a rectangular object in the set, then there is also a white object.

since it is given that every white object is circular, this implies no rectangular object can be white, becoz it cannot be both rectangular and circular, also we can conclude that all rectangular objects are black and thin. Thus if a set contains a rectangular object then it must contain a white object becoz its given not all thin objects are black. that means there are atleast two objects in set if it contains a rectangular object, thus statement (ii) is true

Statement (iii) says Every fat object in the set is circular

Not necessarily since there can be thin circular objects present, thus statement (iii) is false

Kindly correct me if i am wrong !