1. ## Associativity Proof

Prove or disprove in set $\displaystyle \mathbb{R}$ defined algebraic operation $\displaystyle \circ$ associativity if:

$\displaystyle x\circ y = xy - x - y +2$

2. Hello, Bernice!

Prove or disprove in set $\displaystyle \mathbb{R}$ defined algebraic operation $\displaystyle \circ$ associativity if:

$\displaystyle x\circ y \:= \:xy - x - y +2$
I find that it helps to use baby-talk . . .

$\displaystyle x \circ y \:\text{ means: }\:\begin{Bmatrix}\text{multiply the two terms,}\\ \text{minus the first term,} \\ \text{minus the second term,} \\ \text{then add 2.} \end{Bmatrix}$

We will see if: .$\displaystyle (a \circ b)\circ c \:=\: a\circ(b\circ c)$

. . . $\displaystyle a\circ b \;=\;ab - a - b + 2$

$\displaystyle (a\circ b)\circ c \;=\; (ab - a - b + 2)\circ c$

. . . . . . $\displaystyle =\; (ab-a-b+2)c - (ab-a-b+2) - c + 2$

. . . . . . $\displaystyle =\; abc - ac - bc + 2c - ab + a + b - 2 - c + 2$

. . . . . . $\displaystyle =\; {\color{blue}abc - ab - bc - ac + a + b + c}$

. . . $\displaystyle (b\circ c) \;=\;bc - b - c + 2$

$\displaystyle a\circ(b\circ c) \;=\;a\circ(bc - b - c + 2)$

. . . . . . $\displaystyle = \;a(bc-b-c+2) - a - (bc-b-c+2) + 2$

. . . . . . $\displaystyle =\; abc - ab - ac + 2a - a - bc + b + c - 2 + 2$

. . . . . . $\displaystyle =\; {\color{blue}abc - ab - bc - ac + a + b + c}$

They are equal! . . . $\displaystyle (a\circ b)\circ c \;=\;a\circ(b\circ c)$

Therefore, the operation is associative.

3. Beat to it. Nevermind.