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

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Bunuel
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If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  10 Aug 2017, 01:41
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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, \(0 ⊙ y = 2y^2\)


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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  10 Aug 2017, 01:58
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Bunuel wrote:
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, \(0 ⊙ y = 2y^2\)



(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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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  13 Jul 2020, 09:26
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altairahmad wrote:
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.


(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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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  04 Jan 2020, 23:02
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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.
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If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  18 Jun 2020, 09:34
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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.
Show Spoiler

The correct answer is A;
statement 1 alone is sufficient.
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If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  09 Jul 2020, 00:56
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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.
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If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  03 Oct 2021, 09:02
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.
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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  19 Oct 2021, 02:12
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.
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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  19 Oct 2021, 15:58
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faat99 wrote:
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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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  26 Oct 2021, 11:41
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. . .
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Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y? [#permalink] New post  26 Oct 2021, 15:34
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sbolipombo wrote:
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.
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Re: If denotes a mathematical operation, does xy = yx for all x and y? [#permalink] New post  06 Jun 2022, 08:51
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)

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Re: If denotes a mathematical operation, does xy = yx for all x and y? [#permalink] New post  07 Jul 2022, 08:52
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Bunuel wrote:
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

-------------------------------------

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!
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Re: If denotes a mathematical operation, does xy = yx for all x and y? [#permalink] New post  04 Nov 2022, 20:23
Unable to understand this question altogether! What is the "interchangeability" concept that's being used here?
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Re: If denotes a mathematical operation, does xy = yx for all x and y? [#permalink]
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