If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?

Question

If ⊙ denotes a mathematical operation, does  x⊙y = y⊙x  for all x and y?

(1) For all x and y,  x ⊙ y = 2(x^2 + y^2)

(2) For all y,  0 ⊙ y = 2y^2

DS17602.01

Luckisnoexcuse · Accepted Answer

(1) since x and y can be interchanged
Sufficient

(2) if ⊙ denotes + 
then 
0 ⊙ y = 0 + 2y^2
x ⊙ y = x + 2y^2
Here x and y cannot be interchanged
but if 
0 ⊙ y = 2*0^2 + 2y^2
x ⊙ y = 2*x^2 + 2y^2
Here x and y can be interchanged
Not Sufficient

A

KarishmaB · Answer

(2) For all y, 
0⊙y=2y^2

The user defined operator could stand for anything. It is possible that

x⊙y = 2y^2 + x (not symmetrical)
Then 0⊙y = 2y^2

It is also possible that x⊙y = 2y^2 + 2x^2 (symmetrical)
Then also 0⊙y = 2y^2

In one case, x⊙y=y⊙x, in one it is not. Hence not sufficient.

Nagababu436 · Answer

(1) For all x and y, x⊙y=2(x2+y2)

Here X and Y are interchangeable and gives the same value. Hence sufficient.

(2) For all y, 0⊙y=2y2

here y⊙0=0 , the values are not equal if y has values other than 0. Hence insufficient.

coreyander · Answer

OFFICIAL EXPLANATION

1) For all x and y,  x⊙y = 2(x^2 + y^2)  is equal to  y⊙x = 2(y^2 + x^2) ; SUFFICIENT.

2) If  x⊙y = 2(x^2+y^2)  for all x and y, then   0⊙y = 2y^2  for all y and x⊙y = y⊙x for all x and y. However, if  x⊙ y = 2(x^3 + y^2)  for all x and y, then  0⊙y = 2y^2  for all y, but  1⊙2 = 2(1^3) + 2(2^2) = 10  and  2⊙1 
= 2(2^3) + 2(1^2) = 18;  NOT sufficient.
  
The correct answer is A;
statement 1 alone is sufficient.

altairahmad · Answer

chetan2u   VeritasKarishma   GMATBusters

For statement 2, isn't this logic sufficient that the expression only holds true when one of the variables (be it x or y) is 0 whereas the questions asks if it is true for all values of x and y ?

On a related note, is the expression in statement 2 even interchangeable ? I mean the expression hold strictly when x is 0 and y is a variable. Since one of the values is fixed, doesn't that mean that the expression can't be interchanged (i.e x for y and y for as can be done in Statement 1).

Will appreciate some comments to improve my logic here.

faat99 · Answer

OG answer is wrong, answer is D. Welcome anyone proves me otherwise.

The 2nd statement is  SUFFICIENT . If 0⊙y=2y^2 for all y. That means ⊙ cannot be multiple, divide or minus. ⊙ can only me addition.

Therefore back to the statement x⊙y equal to y⊙x for all x and y.

Statement 1 is clear so I won't address it.

faat99 · Answer

Appreciate if someone can clarify why the following does not work:

For the second statement "0*y = 2y^2"
I believe it is possible to identify that * is addition operator:
it cannot be multiplier, ie, for example pick y=100, RHS: 0*y=0, LHS: 2(100)^2 - RHS!=LHS, the relationship does not hold, hence multiplier cannot be the operator. with similar logic, it cannot be divider.
It cannot be minus either, as GMAT does not test imaginary number.
 Therefore * can only be addition, and hence it is sufficient to conclude x+y = y+x

Appreciate any insight.

Basshead · Answer

⊙ isn't restricted to addition/subtraction/multiplication/division -- it's a function.

Since statement 2 doesn't tell us anything about x, we can't conclude that the operation is symmetrical for all x and y.

sbolipombo · Answer

I don't think I understand the interchangeability issue that most people raise here. . .

Would it be enough to say S1 tells us something about all x and y while S2 tells us only about all y YET we are required to determine if the operation applies to all x and y. Therefore S2 is insufficient?

I'm confused. . .

faat99 · Answer

That is correct.
S1 - apply x first or y first doesn't matter.
S2 - no information about x, hence cannot conclude x2+y2 is always true irrespective of order.

Rail · Answer

Hi all, i am confused, which is the right answer to this question? 
Is definitely A? (should be A since official explanation posted above. Still the official explanation on the 2nd statement does not convince me fully)

Thanks

summerbummer · Answer

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I did not understand this question at all. Can someone please help me understand?

Also that, I feel I need to work on such sort of questions. Please if anyone can share a bank of such questions to practice?

Thank you!

NikTek · Answer

Unable to understand this question altogether! What is the "interchangeability" concept that's being used here?

GMATinsight · Answer

Question: does  x⊙y = y⊙x  for all x and y?

To answer this question we need to know what operation is being represented by ⊙

Statement 1  : For all x and y,  x ⊙ y = 2(x^2 + y^2)

i.e. For all x and y,  y ⊙ x = 2(y^2 + x^2)

and both are equal hence 
 SUFFICIENT

Statement 2  : For all y,  0 ⊙ y = 2y^2 
Now, because of 0 we don't know whether first part is being added or first part is added after being squared so mystery about the mathematical operation still persists hence
 NOT SUFFICIENT

Answer: Option A

If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?

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GMAT Data Sufficiency (DS) Questions

If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?

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