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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?

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Math Expert V
Joined: 02 Sep 2009
Posts: 65241
If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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32 00:00

Difficulty:   35% (medium)

Question Stats: 68% (01:23) correct 32% (01:51) wrong based on 749 sessions

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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, $$0 ⊙ y = 2y^2$$

DS17602.01

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Current Student P
Joined: 18 Aug 2016
Posts: 588
Concentration: Strategy, Technology
GMAT 1: 630 Q47 V29 GMAT 2: 740 Q51 V38 Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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6
2
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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##### General Discussion
Intern  B
Joined: 11 May 2019
Posts: 4
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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.
Manager  S
Joined: 17 Jul 2017
Posts: 243
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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

Bunuel can u explain this qsn ,sply 2nd stmnt?
Manager  S
Joined: 17 Jul 2017
Posts: 243
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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1
vanam52923 wrote:
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

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

Can any one explain this question?
Director  V
Joined: 24 Oct 2016
Posts: 711
GMAT 1: 670 Q46 V36 GMAT 2: 690 Q47 V38 Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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vanam52923 wrote:
vanam52923 wrote:
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

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!
Intern  B
Joined: 02 Apr 2020
Posts: 3
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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.
Intern  B
Joined: 24 Jan 2020
Posts: 1
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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

Posted from my mobile device
Manager  S
Joined: 31 May 2020
Posts: 165
If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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.

statement 1 alone is sufficient.
Intern  B
Joined: 15 Jul 2019
Posts: 17
Location: United Kingdom
GMAT 1: 730 Q49 V40
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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

anyone got a PROPER. COMPREHENSIBLE. CLEAR solution for this. none of the posts in the thread deliver
Manager  B
Joined: 27 Mar 2017
Posts: 177
If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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.
Veritas Prep GMAT Instructor V
Joined: 16 Oct 2010
Posts: 10675
Location: Pune, India
Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?  [#permalink]

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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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Karishma
Veritas Prep GMAT Instructor Re: If ⊙ denotes a mathematical operation, does x⊙y = y⊙x for all x and y?   [#permalink] 13 Jul 2020, 08:26

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