# Is the Operation * Associative for All Values of a in Real Numbers?

• lostinmath08
In summary, the conversation discussed the associative law on the set of real numbers and finding the values of a real parameter such that the defined operation is associative. The solution involved substituting values and using the associative law to determine that a must equal 1 for the operation to be associative for all real numbers.
lostinmath08

b1. Homework Statement [/b]

On the set of real numbers R, the following is defined *:RxR arrow R
(x,y) arrow x*y=a(x+y)-xy
find all the values of the real parameter a such that the operation is associative

## Homework Equations

associative law states x*(y*z)=(x*y)*z

## The Attempt at a Solution

x*y = a (x+y) - xy = ax + ay - xy
(m*n) * o = m * (n*o)

(am + an - mn) * o = m * (an + ao - no)
a(am + an - mn) + ao - o (am + an - mn) = am + a (an + ao - no) - m (an + ao - no)
aam + aan - amn + ao - amo - ano + mno = am + aan + aao - ano - amn - amo + mno
am + o = m + ao
am - ao = m - o
a (m - o) = m - o

a = 1 if m does not equal o, and a does not equal 0 if m = 0

im unsure of my solution...any help would be awesome!

lostinmath08 said:
b1. Homework Statement [/b]

On the set of real numbers R, the following is defined *:RxR arrow R
(x,y) arrow x*y=a(x+y)-xy
find all the values of the real parameter a such that the operation is associative

## Homework Equations

associative law states x*(y*z)=(x*y)*z

## The Attempt at a Solution

x*y = a (x+y) - xy = ax + ay - xy
(m*n) * o = m * (n*o)
Why switch to m, n, and o? x, y, and z were working fine!

(Oh, and never use "o" as a symbol for a number- it looks too much like 0 and is too confusing.)

(am + an - mn) * o = m * (an + ao - no)
a(am + an - mn) + ao - o (am + an - mn) = am + a (an + ao - no) - m (an + ao - no)
aam + aan - amn + ao - amo - ano + mno = am + aan + aao - ano - amn - amo + mno
am + o = m + ao
am - ao = m - o
a (m - o) = m - o

a = 1 if m does not equal o, and a does not equal 0 if m = 0

im unsure of my solution...any help would be awesome!
Looks to me like you have it! Remember this a must work for all m! If a must be 1 whenever m does not equal to 0 (and certainly there will are numbers that are not equal to 0!) you had better take a= 1.

As far as "a does not equal 0 if m= 0", I see no problem. 1 is not equal to 0!

would it be wrong to use m, n and p?
also the way i have presented the answer is it legitimate?

No, it's just that after you have written it in terms of x, y, z, I see no reason to change to other symbols.

Yes, just note that in order that your equations be true for all x, y, z, a must be equal to 1.

Thanks so much for your help!

## 1. What is the Associative Law?

The Associative Law is a mathematical rule that states that the grouping of numbers in an operation does not change the result. In other words, when adding or multiplying three or more numbers, the order in which the numbers are grouped does not affect the final answer.

## 2. How do I use the Associative Law?

To use the Associative Law, simply group the numbers in the operation in any way you like. For example, if you are adding 2+3+4, you can group it as (2+3)+4 or 2+(3+4), and the result will be the same.

## 3. Is the Associative Law only applicable to addition and multiplication?

No, the Associative Law also applies to subtraction and division. For example, (8-4)-2 is equivalent to 8-(4-2).

## 4. Can I use the Associative Law for all numbers?

Yes, the Associative Law can be applied to any numbers, including whole numbers, fractions, and decimals. It can also be used for variables in algebraic expressions.

## 5. Why is the Associative Law important?

The Associative Law is important because it allows us to simplify complex mathematical expressions and equations. It also helps us to better understand the properties of numbers and how they behave in mathematical operations.

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