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Commutator

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Commutator

by Sher1 » Thu Aug 06, 2009 9:20 pm
For any operation ? that acts on two numbers x and y, the commutator is defined as x?y – y?x. For which of the following operations is the commutator not equal to zero for some values of x and y?

I. x?y = xy
II. x?y = (x – y)^2
III. x?y = x^3 + 3x^2y + 3xy^2 + y^3

(A) I only
(B) II only
(C) III only
(D) II and III only
(E) I, II, and III
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by real2008 » Thu Aug 06, 2009 9:30 pm
I don't find the answer among the choices available
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by aspiregmat » Thu Aug 06, 2009 9:31 pm
I x?y = xy

so commutator = xy - yx so always 0

II x?y = (x-y)^2

so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0

III x?y = (x+y)^3

so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0

so IMO ans = C

OA Plz.
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by real2008 » Thu Aug 06, 2009 9:35 pm
aspiregmat wrote:I x?y = xy

so commutator = xy - yx so always 0

II x?y = (x-y)^2

so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0

III x?y = (x+y)^3

so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0

so IMO ans = C

OA Plz.
can u pl. explain how do u get the highlighted term? I mean why it is (y-x)^3 and not (y+x)^3
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by aspiregmat » Thu Aug 06, 2009 9:41 pm
real2008 wrote:
aspiregmat wrote:I x?y = xy

so commutator = xy - yx so always 0

II x?y = (x-y)^2

so commutator = (x-y)^2 - (y-x)^2 taking -1 in one and squaring again makes it 1 so always 0

III x?y = (x+y)^3

so commutator = (x+y)^3 - (y-x)^3 its cube so -1 common wont help so not always 0

so IMO ans = C

OA Plz.
can u pl. explain how do u get the highlighted term? I mean why it is (y-x)^3 and not (y+x)^3
My mistake ...

it should be (x+y)^3 - (y+x)^3 so Ans IMO none of the above :) , which is not in the ans choice....
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by Sher1 » Fri Aug 07, 2009 11:50 am
I also thought the answers should be none. This is a Manhattan Gmat challenge question I found on MBA mission. Below is the exp if anyone can make sense of it please share.


https://www.mbamission.com/blog/2009/08/ ... llenge-63/
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