(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.
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(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
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(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.

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.

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.

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.

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.

⊙ 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.
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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. . .

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.

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