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if the operation & is defined for all integers a and b

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Manager
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if the operation & is defined for all integers a and b [#permalink] New post 08 Nov 2005, 21:22
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

What's the easiest way to approach this to figure out if III holds true? you could substitue numbers once to see if it works out, but to double or even triple check is incredibly time consuming and could get hard to keep track of your work.
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 [#permalink] New post 09 Nov 2005, 18:23
Simplyfying III is not too difficult -

(a & b) & c = [a+b-ab]&c = a+b-ab+c-ac-bc+abc
=a+b+c-ab-bc-ac+abc ---(1)

a & (b & c) = a&[b+c-bc] = a+b+c-bc-ab-ac+abc
=a+b+c-ab-bc-ac+abc ---(2)

As we can see (1) = (2). Hence III is true.
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Re: Operation function [#permalink] New post 09 Nov 2005, 21:39
Jennif102 wrote:
What's the easiest way to approach this to figure out if III holds true? you could substitue numbers once to see if it works out, but to double or even triple check is incredibly time consuming and could get hard to keep track of your work.


you can substitute too but after all you have to again re-substitute those values again.

E. all choices are possible.
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Re: Operation function [#permalink] New post 09 Nov 2005, 21:45
The answer is E.

I am not sure if there is an easy way to evaluate III.
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 [#permalink] New post 04 Dec 2006, 18:10
Could someone please go through all the steps and solve
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 [#permalink] New post 04 Dec 2006, 19:20
I get the same problem in Gmatprep last weekend.
I would suggest you work them out and be 100% sure of the answer. It is safer to work out such 'unique operation' ,since we don't know what would happen if we simplify the steps.
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 [#permalink] New post 05 Dec 2006, 14:50
I would approach this problem by identifying if there's "symmetry" in the function.

Symmetry here means that if you substitute a for b then the function remains the same.

In this case
a&b = a+b-ab
b&a = b+ a -ba

This is a symmetric function, so I would have expected III to be true too.
  [#permalink] 05 Dec 2006, 14:50
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