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.
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You can always test values that's an alternative

Let a=1,b=2 and c=3

Definition => a@b=a+b-ab

Option 3

III. (a@b)@c = a@(b@c)

a@b = 1 + 2 -2 = 1
1@3 = 1+3 -3 = 1

B@c = 2@3 = 5-6 = -1
1@-1 = 1 - 1 - ( -1 * 1)
=1

LHS = RHS

(I) and (II) are obviously correct.

For (III)

(a+b-ab)@c = (a + b - ab)@c = a + b - ab + c - c(a + b - ab) = a + b - ab + c - ac - ab + abc

a@(b@c) = a@(b + c - bc) = a + b + c - bc - a(b + c - bc) = a + b + c - bc -ab - ac + abc

Answer - E
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Sorry for bumping up an old thread, I have a doubt: my approach for solving the question was to assume that the operation in this case was the union between two sets, a and b, and consequently the three points were the properties of the union of sets. Is that a correct approach or might it be too risky in the actual exam? (or is it even a wrong assumption, and I got it right out of luck?)

I. a@b = b@a
\(a+b-ab=b+a-ab\) TRUE!

II. a@0 = a
\(a+0-0 = a\) TRUE!

III. (a@b)@c = a@(b@c)
\(a+b-ab+c-ac-bc+abc = b+c-bc + a - ab-ac+abc\) STRIKE OUT DUPLICATES ON RHS and LHS! TRUE!

Answer: I,II, and III or (E)

That's not correct. Stem defines some function @ for all integers a and b by a@b=a+b-ab. For example if a=1 and b=2, then a@b=1@2=1+2-1*2.

Hope it's clear.
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in equ. 3 :

I put three random numbers like (5,3,2) and tested it. however there's always a little chance of error.

Been looking at this for a while and still can't figure it out.. I thought that we must ALWAYS first do the calculations in the brackets and open them, and then do the remaining calculations. as we have a@(b@c), how come do you straight come up to (a + b - ab)@c, when it's b@c in the brackets, not a@b anymore.. finding this one a bit confusing.. thanks for explaining in advance

EDIT:

OK, I think I get it now.. pls, have a look at my upload and let me know if I am correct.. this is the left side of the equation in (III). With the right one, we do the exact same thing, right?

Hey there folks, sorry to bump on an old thread. Just wondering, is there a way to evaluate statement 3 faster?
I believe this questions takes around 2 minutes and evaluating statement 3 takes a lot of work and is prone to errors.

Just wondering if I'm doing this the correct/most efficient way

Thanks guys
Cheers!

J

Would it be smart to pick numbers for each of the variables? I selected 1&3 only since I got mixed up with the letter variables. Is picking number the most efficient way to approach this problem?

In this problem we have been asked to check the commutative and associative property of the given function. These properties are defined as below:

Commutative: In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

Associative: Within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not changed. That is, rearranging the parentheses in such an expression will not change its value.

If you're wondering if commutativity implies associativity in mathematics then the answer is NO. However, for simple addition and multiplication functions commutativity does imply associativity and hence in such cases option 3 need not be tested if option 1 is true. However, the only way to solve such problems which involve functions other than simple addition and multiplication would be to solve the expression completely as stated above.

Well, can't argue that learning the definitions is in fact quite interesting and thank you for that.
Nevertheless, I was really intereted in solving statement 3 quicker/faster/more efficient
So, being able to recognize if operations in the different order given will yield same result without having to go through all the long distribution process.

I will try to come up with a faster way but if anyone else come's up with a nice and elegant approach I'd be happy to give some nice Kudos for the collection

Cheers
J

Just took this question today, and I was also wondering if there were a way to solve it more quickly than performing the heavy manipulations that are in III or guessing numbers.

Hi Bunuel,

Here in second statement how you are getting 0=0?

Thanks.

II says: \(a@0 = a\)

LHS = \(a@0=a+0-a*0=a\).
RHS = a

a=a.
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I have a question regarding this problem. Don't we have to test for negative numbers while expanding the equation? This is where I got stuck and it became time consuming for me at which point I guessed and moved on.
Please advise why that's not a factor.

Yes, tested for negative values and got stuck. For a@0 = a. If we put a is negative then it doesn't hold. Any thoughts on this? Where am I going wrong?

It does: lets say a=-4 ---> a@0=-4@0 = -4+0-0*-4 = -4 =a. This HAS to be true even by the underlying algebra!

Thus satisfies the given equation.

Can you show your steps for calculations?
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You are right, taking negative values I jumbled up equations. Thank you for your help.