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Math Expert V
Joined: 02 Sep 2009
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If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Question Stats: 68% (01:06) correct 32% (01:07) wrong based on 860 sessions

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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

Problem Solving
Question: 18
Category: Arithmetic Properties of numbers
Page: 64
Difficulty: 600

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Math Expert V
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Posts: 58465
If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

The number of integers that belong to set X ONLY is 10-6=4;
The number of integers that belong to set Y ONLY is 18-6=12;

The number of integers that belong to set X or set Y, but not both is 4+12=16.

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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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1
IMO B.

Set X=10
Set Y=18

both X&Y = 6

(Either X or Y or both) = (X) + (Y) - (both X&Y) = 10+18-6 = 22

Now we want a set of integers from either X or Y but not from both X and Y
X@Y = (Either X or Y or both) - (Both X&Y) = 22-6 = 16.
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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

As per Set theory :
A@B= A + B - 2(A n B), so 10 + 18-2*6 = 16
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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174
Attachment: untitled1.PNG [ 3.39 KiB | Viewed 11247 times ]

Sol: Look at above figure.
Now X@Y = Number of elements in X and Y which are not present in Both.

So X@Y= 10-6+18-6= 16 Ans B
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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

Exactly 1 = X + Y - 2(X&Y)

When you add X and Y the intersection gets added twice hence we have to deduct it twice Exactly 1 = 10 + 18 - 12 = 16

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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Bunuel wrote:
carcass wrote:
If X and Y are sets of integers, X # Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X # Y consists of how many integers?

A. 6
B. 16
C. 22
D. 30
E. 174

The number of integers that belong to set X ONLY is 10-6=4;
The number of integers that belong to set Y ONLY is 18-6=12;

The number of integers that belong to set X or set Y, but not both is 4+12=16.

Hi

I know its silly question but can you please clear my understanding?
Why cannot I do.... 10+18-6??? am I not deducting both from X and Y by doing this??
Math Expert V
Joined: 02 Sep 2009
Posts: 58465
Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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GGMAT760 wrote:
Bunuel wrote:
carcass wrote:
If X and Y are sets of integers, X # Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X # Y consists of how many integers?

A. 6
B. 16
C. 22
D. 30
E. 174

The number of integers that belong to set X ONLY is 10-6=4;
The number of integers that belong to set Y ONLY is 18-6=12;

The number of integers that belong to set X or set Y, but not both is 4+12=16.

Hi

I know its silly question but can you please clear my understanding?
Why cannot I do.... 10+18-6??? am I not deducting both from X and Y by doing this??

That way you'll get the total number of elements in X and y, while we need the number of elements that belong to set X or set Y, but not both.

6 elements belong to both X and y, thus there are 10-6=4 unique elements in X and 18-6=12 unique elements in Y. Thus there are total of 4 + 12 = 16 unique elements in X and Y.

Hope it's clear.
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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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X #Y represents the shaded region as shown in diagram

Answer = 16 = B
Attachments x.png [ 5.24 KiB | Viewed 5857 times ]

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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Is it possible to solve this problem using a matrix?
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Joined: 04 Jan 2015
Posts: 3092
Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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cg0588 wrote:
Is it possible to solve this problem using a matrix?

Hi cg0588,

The question asks us the number of integers which belong to set X or Set Y but not both. This would be equal to the number of integers which belong to only set X + number of integers which belong to only set Y

Please find below the matrix diagram of the solution We are given that set X consists of 10 integers out of which there are 6 integers which are common to set Y. Hence integers which belong to only set X = 10 - 6 = 4

Similarly, we know that set Y consists of 18 integers. As there are 6 integers which are common to set X, we will have 18 - 6 = 12 integers which belong to only set Y.

Thus number of integers which belong to set X or set Y but not both = 4 + 12 = 16

Hope it's clear Regards
Harsh
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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Quote:

If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

Note that the 6 numbers belonging to both sets must be subtracted from set X and again from set Y.

We can use the equation:

X@Y= set X - both + set Y - both

X@Y = 10 - 6 + 18 - 6 = 16

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Re: If X and Y are sets of integers, X@Y denotes the set of inte  [#permalink]

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Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

If X and Y are sets of integers, X@Y denotes the set of integers that belong to set X or set Y, but not both. If X consists of 10 integers, Y consists of 18 integers, and 6 of the integers are in both X and Y, then X@Y consists of how many integers?

(A) 6
(B) 16
(C) 22
(D) 30
(E) 174

Problem Solving
Question: 18
Category: Arithmetic Properties of numbers
Page: 64
Difficulty: 600

(10-6) + (20-6) = 16 Re: If X and Y are sets of integers, X@Y denotes the set of inte   [#permalink] 22 Feb 2019, 14:44
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