New Operator $$\odot$$

Exam style question: something not seen before

So this is a scary looking question! It introduces a symbol you might never have seen before, language you might not recognise and it could be hard to know where to start. Here's a step by step approach to each part of the question, explaining what needs to be done and why.

Question:

For all numbers a and b, let $$a\odot b$$ be defined by $a\odot b = ab + a + b$ For all numbers x, y, and z, which of the following must be true?
I. $$x\odot y = y\odot x$$
II. $$(x - 1)\odot (x + 1)=(x\odot x) - 1$$
III. $$x\odot(y + z) = (x\odot y) + (x \odot z)$$

The $$\odot$$ symbol is like a new function. We know what to do when we see $$6\times5$$ or $$6+5$$ but what do we do when we see $$6\odot5$$?? Well the question tells us; $a\odot b = ab + a + b$ $6\odot5 = 6\times 5 + 6 + 5 = 41$
I. $x\odot y = xy + x + y = yx + y + x = y\odot x$ so I. is TRUE
II. \begin{align} (x - 1)\odot(x + 1) &= (x - 1)(x + 1) + (x - 1) + (x + 1) \\[8pt]&= x^2 -1 + 2x \\[8pt]&= x^2 + x + x - 1 \\[8pt]&= x\odot x - 1 \end{align} so II. is TRUE
The next two lines look at each side of the equation in part III individually. We're seeing if they're the same, so first we get rid of the new operator by using its definition, then comparing the two new expressions. $x\odot(y + z) = x(y + z) + x + (y + z) = xy + xz + x + y + z$ $(x\odot y) + (x\odot z) = xy + x + y + xz + x + z = xy + xz + 2x + y + z$ Now let's compare these two right hand sides. $xy + xz + x + y + z \neq xy + xz + 2x + y + z$ $x\odot(y + z) \neq (x\odot y) + (x\odot z)$ So III. is NOT TRUE.