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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2506
Given Kudos: 1055
GPA: 3.39
If the operation @ is defined for all integers a and b by a@b = a + b
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16 Nov 2021, 08:16
16 Nov 2021, 08:16
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Expert Reply
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Question Stats:
63% (01:54) correct
36% (02:31) wrong based on 11 sessions
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If the operation \(@\) is defined for all integers a and b by a@b = a + b - ab, which of the following statements must be true for all integers a, b and c?
I. a@b = b@a
II. a@0 = a
III. (a@b)@c = a@(b@c)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II and III
I. a@b = b@a
II. a@0 = a
III. (a@b)@c = a@(b@c)
(A) I only
(B) II only
(C) I and II only
(D) I and III only
(E) I, II and III
Re: If the operation @ is defined for all integers a and b by a@b = a + b
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24 Nov 2021, 10:41
24 Nov 2021, 10:41
1
Expert Reply
We have that: \(a@b=a+b-ab\)
I. \(a@b = b@a\) --> \(a@b=a+b-ab\) and \(b@a=b+a-ab\) --> \(a+b-ab=b+a-ab\), results match;
II. \(a@0 = a\) --> \(a@0=a+0-a*0=0\) --> \(0=0\), results match;
III. \((a@b)@c = a@(b@c)\) --> \((a@b)@c=a@b+c-(a@b)*c=(a+b-ab)+c-(a+b-ab)c=a+b+c-ab-ac-bc+abc\) and \(a@(b@c)=a+b@c-a*(b@c)=a+(b+c-bc)-(b+c-bc)a=a+b+c-bc-ab-ac+abc\), results match.
Answer: E.
I. \(a@b = b@a\) --> \(a@b=a+b-ab\) and \(b@a=b+a-ab\) --> \(a+b-ab=b+a-ab\), results match;
II. \(a@0 = a\) --> \(a@0=a+0-a*0=0\) --> \(0=0\), results match;
III. \((a@b)@c = a@(b@c)\) --> \((a@b)@c=a@b+c-(a@b)*c=(a+b-ab)+c-(a+b-ab)c=a+b+c-ab-ac-bc+abc\) and \(a@(b@c)=a+b@c-a*(b@c)=a+(b+c-bc)-(b+c-bc)a=a+b+c-bc-ab-ac+abc\), results match.
Answer: E.
Re: If the operation @ is defined for all integers a and b by a@b = a + b
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07 Mar 2023, 13:42
07 Mar 2023, 13:42
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Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).
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