When is this operation associative?

Algebra Level 3

Suppose ⊛ \circledast is an operation defined on the real numbers as follows:

x ⊛ y = 2 x + y . x \circledast y = 2x + y.

When is the equation

( x ⊛ y ) ⊛ z = x ⊛ ( y ⊛ z ) (x \circledast y) \circledast z = x \circledast (y \circledast z) true?

Only when x = 0 x=0 Never True Always True Only when z = 0 z=0

Damon Demas by Damon Demas

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1 solution

We have:

( x ⊛ y ) ⊛ z = 2 ( x ⊛ y ) + z = 2 ( 2 x + y ) + z = 4 x + 2 y + z \left(x\circledast y\right)\circledast z=2(x\circledast y)+z=2(2x+y)+z=4x+2y+z

x ⊛ ( y ⊛ z ) = 2 x + ( y ⊛ z ) = 2 x + ( 2 y + z ) = 2 x + 2 y + z x\circledast\left(y\circledast z\right)=2x+\left(y\circledast z\right)=2x+(2y+z)=2x+2y+z

So ( x ⊛ y ) ⊛ z = x ⊛ ( y ⊛ z ) ⇔ 4 x + 2 y + z = 2 x + 2 y + z ⇔ x = 0 \left(x\circledast y\right)\circledast z=x\circledast\left(y\circledast z\right) \Leftrightarrow 4x+2y+z=2x+2y+z\Leftrightarrow \boxed{x=0} .

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