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
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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.

Can any one explain this question?

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.

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.