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. $x\odot (x+z)=(x\odot y)+(x\odot z)$

• 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

The ⊙ symbol is like a new function. We know what to do when we see "6*5" or "6+5" but what do we do when we see "6⊙5"?

Well the question tells us;

a⊙b = ab + a + b

So $6\odot 5=6\times 5+6+5=41$.

(I) $x\odot y=xy+x+y=yx+y+x=y\odot x$

so I. is true

(II)$(x-1)\odot (x+1)=(x-1)\times (x+1)+(x-1)+(x+1)=x^2-1+2x=x^2+x+x-1=x\odot x-1$

so II. is ture

(III)$x\odot (y+z)=x(y+z)+x+y+z=xy+xz+x+y+z$

$(x\odot y)+(x\odot z)=xy+x+y+xz+x+z=xy+xz+2x+y+z$

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

So III. is not true

Then II and III option is true