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

Let $E\ =\ \{ \ 1,2,3,4,5,6,7,8,9,10\ \},\ F\ =\ \{\ 5,10\ \},\ G\ =\ \{\ 2,4,6,8,10\ \}\\E\ \cap\ F\ =\ \{\ 5,10\ \}\\F\ \cap\ G\ =\ \{\ 10\ \}\\X\ =\ \{\ 5\ \}\ \\E\ \cap\ G\ =\ \{\ 2,4,6,8,10\ \}\\E\ -\ E\ \cap\ G\ =\ \{ \ 1,3,5,7,9\ \}\\E\ -\ F\ =\ \{\ 1,2,3,4,6,7,8,9\}\\Y\ =\ (E\ -\ (E\ \cap\ G\ ))\ -\ (E\ -\ F) =\ \{\ 5 \ \}\\ \therefore\ X\ = \ Y. So\ answer\ is\ (C)$

Note that this example also eliminates all the other options.

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For these type of questions, this method seems simple and fast

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Answer 1

 

 

$X = Y$

Option C is answer.

Answer 2

hope it might help.....

Comments

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

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.