GATE CSE 2006 | Question: 22

Question

Let $E, F$ and $G$ be finite sets. Let

• $X = (E ∩ F) - (F ∩ G)$ and
• $Y = (E - (E ∩ G)) - (E - F)$.

Which one of the following is true?

• A. $X ⊂ Y$
• B. $X ⊃ Y$
• C. $X = Y$
• D. $X - Y ≠ \emptyset$ and $Y - X ≠ \emptyset$

Comments

Try to convert into boolean it will be much easier..

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@Aadhyaa Gupttaa Can you explain ur approach please?

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you can check this answer:

https://gateoverflow.in/983/gate-cse-2006-question-22?show=411544#a411544

 

Answer 1

 

 

$X = Y$

Option C is answer.

Comments

Wash sir samjhane ka tarika thoda jyada casual hai

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Solve using boolean algebra also

 

Answer 2

hope it might help.....

Comments

I find this as one of the best and the easiest method.

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same here

Answer 3

Answer is $C$ using Venn diagram.

Answer 4

Let E,F, and G belongs to same universe of discourse U, then we can write  E-F=E ∩ F' =EF' .

X = (E∩F) - (F∩G)  = (E∩F) ∩ (F∩G)' =EF (F' + G') =EFG' =(E ∩F ∩G' )

Y=(E−(E∩G))−(E−F) =E (EG)' - (EF') = E(E'+G') - (EF') = EG' - EF'= EG' (EF')' = EG'(E'+F) =EFG' = (E∩F∩G')

We can clearly see that ,X=Y.

Option (C) X=Y   is the correct answer.

Answer 5

X=(E∩F)−(F∩G)
Y=(E−(E∩G))−(E−F)
Let E={1,2,3,4,5}   positive integers

F={2,3,5,7}  prime numbers

G={1,3,5} odd numbers

E∩ F ={2,3,5} ,F∩G={3,5}   so X={2}

E∩G={1,3,5} ,E-{E∩G}={2,4},E-F={1,4}  

Y={2} so X and Y are same so option C is right

Answer 6

BEST METHOD 

the solution can be obtained for boolean algebra as follows:
X=(E∩F)−(F∩G)
=EF−FG
=EF∩(FG)′
=EF.(F′+G′)
=EFF′+EFG′
=EFG′
 

 

Similarly, Y=(E−(E∩G))−(E−F)
=(E−EG)−(E.F′)
=E.(EG)′−EF′
=E.(E′+G′)−EF′
=EG′−EF′
=EG′.(EF′)′
=EG′.(E′+F)
=EE′G′+EFG′
=EFG′
 

Therefore, X=Y