# Given x*y=xy-2x-2y+6, x*2=2, calculate -2009*-2008*-----*-1*0*1*----2009?

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You should use the information provided by the problem `x*2 = 2` , hence, you may group the terms around the term 2 such that:

`((-2009)*......*(-1)*0*1)*2*(3*4*...*2009)`

Reasoning by analogy yields that `((-2009)*......*(-1)*0*1)*2 = 2.`

You should prove that the given law of composition is associative, hence, it satisfies the associative law such that:

`(x*y)*z= x*(x*z)`

You need to use the following form of the law of composition, such that:

`x*y = (x-2)(y-2) + 2`

`(x*y)*z = ((x-2)(y-2) + 2)*z = ((x-2)(y-2) + 2 - 2)(z-2) + 2`

`(x*y)*z = (x-2)(y-2)(z-2) + 2`

`y*z = (y-2)(z-2) + 2`

`x*(y*z) = x*((y-2)(z-2) + 2)`

`x*(y*z) = (x-2)((y-2)(z-2) + 2 - 2) + 2`

You need to evaluate if `(x*y)*z = x*(y*z)`  such that:

`(x-2)(y-2)(z-2) + 2 = (x-2)((y-2)(z-2)) + 2`

Reducing 2 both sides yields:

`(x-2)(y-2)(z-2) = (x-2)(y-2)(z-2)`

Since the law is associative and commutative, hence `x*2 = 2*x = 2.`

Reasoning by analogy yields:

`2*(3*4*...*2009) = 2`

Hence, evaluating the expression `((-2009)*......*(-1)*0*1)*2*(3*4*...*2009), ` using the information provided by the problem, yields `((-2009)*......*(-1)*0*1)*2*(3*4*...*2009) = 2.`

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