Question

Given that x*y=1+x+y, show wheather the binary operation is associative algebrically

Answer 1

A binary operation $\cdot$ is associative when $(x \cdot y) \cdot z = x \cdot (y \cdot z)$ for any $x, y, z$ .

Suppose we have

$(x \cdot y) \cdot z$

We can replace the binary operation with addition using its definition:

$(x \cdot y) \cdot z = (1 + x + y) \cdot z = 1 + (1+x+y) +z$

Now we can remove parentheses (using associativity of addition), regroup terms (using commutativity of addition) and put parentheses where we want (using associativity of addition once again):

$1 + (1 + x + y) + z = \\ =1 + 1 + x + y + z = \\ = 1 + x + 1+y +z = \\ =1 + x + (1+y+z)$

$1+y+z = y \cdot z$ by definition of this binary operation.

$1+ x + (1+y+ z) = 1+x+(y \cdot z)$

$1+x+(y \cdot z) = x \cdot (y \cdot z)$ by definition of our binary operation.

We proved that $(x \cdot y) \cdot z = x\cdot (y \cdot z)$ which means we proved that the binary operation is associative algebraically. What was to be shown.