GATE CSE 2005 | Question: 8

Question

Let $A, B$ and $C$ be non-empty sets and let $X = ( A - B ) - C$ and $Y = ( A - C ) - ( B - C ).$ Which one of the following is TRUE?

• A. $X = Y$
• B. $X ⊂ Y$
• C. $Y ⊂ X$
• D. None of these

Comments

A?

 

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Convert into boolean you will easily get the answer..

also, you need to keep in mind this expression

A-B= AB'

 

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Intutive Answer  

Correct answer  is:

$$\boxed{\text{A. } X = Y}$$

Simple proof : 

$$A - B = A \setminus B$$

means “elements in $A$ but not in $B$.”

So,

$$X = (A - B) - C$$

 elements that are in $A$, not in $B$, and not in $C$.

So,

$$X = A \setminus (B \cup C)$$

Now look at:

$$Y = (A - C) - (B - C)$$

Now ,

Take any element $x$.

For $x \in X$:

$$x \in (A - B) - C$$

means:

• $x \in A$

• $x \notin B$

• $x \notin C$

So $x$ satisfies all three conditions.

For $x \in Y$:

$$x \in (A - C) - (B - C)$$

means:

• $x \in A$

• $x \notin C$

• and $x \notin (B - C)$

Now $x \notin (B - C)$ means:

$$\text{not}(x \in B \text{ and } x \notin C)$$

But since we already know $x \notin C$, this simplifies to:

$$x \notin B$$

So again we get:

• $x \in A$

• $x \notin B$

• $x \notin C$

Thus $x \in X$.

Since every element of $Y$ is in $X$, and every element of $X$ is in $Y$. Hence,

$$X = Y$$

Final answer

$$\boxed{X = Y}$$

So the correct option is:

$$\boxed{\text{A}}$$

Answer 1

 

$X = (A-B) - C = {1,4} - {4,5,6,7 } = 1$

$Y = (A-C) -(B-C) = {1,2} - {2,3} = 1$

So, $X = Y$.

Answer $A$

Comments

Best way

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I have also solved this in the same way.

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Plz remove this answer from best category and convert into simple answer ..Best answer should be based on Proof , intution and better explanation ....50% math answer are selected as best answer which not even good answer ...@admin plz look into this

Answer 2

X=(A-B)-C

=>AB' - C

=>AB'C'

 

Y=(A-C)-(B-C)

=>AC'-BC'

=>AC'(B+C')

=>AC'B'

 

so X=Y

note:P-Q is equivalent to P intersection Q'

or P-Q=PQ'

Comments

This should be the accepted answer. Also, there's a slight mistake. It should be $AC'(B'+C)$ instead of $AC'(B+C')$ in the second-last step of Y

Answer 3

From the venn diagram it is clear that, X=Y (Option A)

Answer 4

we can also check by taking any three random sets

Answer 5

X=(A−B)−C
=(A∩B′)−C
=(A∩B′)∩C′
=AB′C′
 

Y=(A−C)−(B−C)
=(A∩C′)−(B∩C′)
=(AC′)−(BC′)
=(AC′)∩(BC′)′
=(AC′)∩(B′+C)
=(AC′).(B′+C)
=AC′B′+AC′C
=AC′B′ (since C′C=1)
=AB′C′ (commutative property)
∴X=Y

it is clear that, X=Y (Option A)

 

😊😊😊😊😊😊😊😊😊😊😊😊

Answer 6

https://docs.google.com/document/d/1YflPrYyj-GLEwhu34fy49gBY65LqT5La86ZPi6moQaM/edit?usp=sharing

Here i Shared Both method teach by @Deepak Poonia sir