Question

For all numbers a and b, let $a⨀b$ be defined by $a⨀b=ab+a+b$. Then for the numbers $x$, $y$ and $z$, which of the following is/are true?
I. $x⨀y=y⨀x$
II. $(x-1)⨀(x+1)=(x⨀x)-1$
III. $x⨀(y+z)=(x⨀y)+(x⨀z)$

• A. I only
• B. II only
• C. III only
• D. I and II only
• E. I, II, and III

Answer 1

The correct option is D I and II only

$For\left(I\right)$

$x\odot y=y\odot x$

$x\odot y=xy+x+y\Rightarrow x\odot y=y\odot x$

$y\odot x=yx+y+x$

$For\left(II\right)$

$(x-1)\odot (x+1)$

$=(x-1)(x+1)+x-1+x+1$

$=x^2-1+2x$

$(x\odot x)-1=x\times x+x+x-1$

$x^2+2x-1$

$\therefore (x-1)\odot (x+1)=(x\odot x)-1$

$For\left(III\right)$

$x\odot (y+z)$

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

$=xy+xz+x+y+z$

$x\odot y=xy+x+y$

$x\odot z=xz+x+z$

$x\odot y+x\odot z=xy+xz+x+y+x+z$

$=xy+xz+2x+y+z$

$\Rightarrow x\odot (y+z)\ne x\odot y+x\odot z$

$\therefore \left(I\right)\&\left(II\right)$ are true.