Associative Operation/Examples/Non-Associative/xy+1

Example of Non-Associative Operations

Let $\R$ denote the set of real numbers.

Let $\circ$ denote the operation on $\R$ defined as:

$\forall x, y \in \R: x \circ y := x y + 1$

Then $\circ$ is not an associative operation, despite being commutative.

Let $x, y, z \in \R$.

We have:

$\ds \paren {x \circ y} \circ z$

$=$

$\ds \paren {x y + 1} z + 1$

$\ds $

$=$

$\ds x y z + z + 1$

$\ds x \circ \paren {y \circ z}$

$=$

$\ds x \paren {y z + 1} + 1$

$\ds $

$=$

$\ds x y z + x + 1$

But unless $x = z$ it is not the case that $\paren {x y + 1} z + 1 = x y z + x + 1$.

However, $x y + 1 = y x + 1$ and so $\circ$ is commutative.

$\blacksquare$

1965: J.A. Green: Sets and Groups ... (previous) ... (next): Chapter $4$. Groups: Exercise $2$