(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.
_________________

(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
_________________

(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.

Bunuel can u explain this qsn ,sply 2nd stmnt?

Can any one explain this question?

VeritasKarishma Bunuel Can you please explain the solution for this question? Thanks!

Could we replace "operation" by "function" here ? I cannot understand what's going on.

I understand that if we have x o y = 2(x^2+y^2) we can change the order and come up with x o y = y o x. But which of "+", "-", "/" and "*" could work with "all x and all y". I cannot understand why we use an equal sign.

For st 2, we can see that 0 to any nonzero power = 0 and thus 0•2y^2=2y^2
Thus when we replace 0 with x, then x can technically take any power.

However, when x takes any power, then sometimes it CANNOT be interchanged with y eg. 2(X^3+y^2)≠2(y^3+x^2) for say y=2 and x=3 [ie. 2(27+4)≠2(8+9)]

But sometimes it CAN be interchanged with y^2, ie. When x has the power 2. Not Sufficient

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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.

anyone got a PROPER. COMPREHENSIBLE. CLEAR solution for this. none of the posts in the thread deliver

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.

Where are they when you need them? chetan2u VeritasKarishma GMATBusters bunuel

Hello BrentGMATPrepNow,

I was wondering whether you could provide a solution for this one, I don't understand how I should proceed,
for instance for statement 1 I thought that I had to test some numbers eg. x=1 , y=2 and see which mathematical operation does the sign represent but I think that's not the point because there is not a mathematical operation(+-*...) that could get us to x⊙y=2(x2+y2)