Question

For all the numbers a and b, let $a\odot b$ be defined by $a\odot b=ab+a+b$. Then for the numbers x, y and z, which of the following is/are true?
(i) $x\odot y=y\odot x$
(ii) $\left( x-1 \right)\odot \left( x+1 \right)=\left( x\odot x \right)-1$
(iii) $x\odot \left( y+z \right)=\left( x\odot y \right)+\left( x\odot z \right)$

Answer 1

The definition of the special operator for any two numbers a and b is given to us. To verify which of the given statements are true, we need to take each statement and define the operator on both the sides of the equal to sign. If the left hand side is equal to the right hand side, the statement is true. If they are not equal, the statement is false.

Complete step-by-step answer:
It is given to us that for any two numbers a and b, $a\odot b$ is defined as $a\odot b=ab+a+b$.
The first statement given to us is $x\odot y=y\odot x$
First of all, we will define the left hand side.
$\Rightarrow x\odot y$ = xy + x + y
Now, let us see the right hand side.
$\Rightarrow y\odot x$ = yx + y + x
$\Rightarrow y\odot x$ = xy + x + y
But we already defined that $x\odot y$ = xy + x + y.
$\Rightarrow x\odot y=y\odot x$
Therefore, the given statement is true.
The second statement given to us is $\left( x-1 \right)\odot \left( x+1 \right)=\left( x\odot x \right)-1$
First of all, we will define the left hand side.
$\Rightarrow \left( x-1 \right)\odot \left( x+1 \right)$ = (x – 1)(x + 1) + (x – 1) + (x + 1)
We will apply the rule $\left( a-b \right)\left( a+b \right)={{a}^{2}}-{{b}^{2}}$.
$\Rightarrow \left( x-1 \right)\odot \left( x+1 \right)={{x}^{2}}+2x-1$
Now, we shall proceed to the right hand side.
$\begin{aligned} & \Rightarrow \left( x\odot x \right)-1=x\times x+x+x-1 \\\ & \Rightarrow \left( x\odot x \right)-1={{x}^{2}}+2x-1 \\\ \end{aligned}$
But we already proved that $\left( x-1 \right)\odot \left( x+1 \right)={{x}^{2}}+2x-1$.
$\Rightarrow \left( x-1 \right)\odot \left( x+1 \right)=\left( x\odot x \right)-1$
Therefore, the given statement is true.
The third statement given to us is $x\odot \left( y+z \right)=\left( x\odot y \right)+\left( x\odot z \right)$
First of all, we will define the left hand side.
$\Rightarrow x\odot \left( y+z \right)$ = x(y + z) + x + (y + z)
$\Rightarrow x\odot \left( y+z \right)$ = xy + xz + x + y + z
Now, we shall proceed to the right hand side.
$\Rightarrow \left( x\odot y \right)+\left( x\odot z \right)$ = xy + x + y + xz + x + z
$\Rightarrow \left( x\odot y \right)+\left( x\odot z \right)$ = xy + xz + 2x + y + z
Since the left hand side is not equal to the right hand side, the given statement is not true.
Hence, statements (i) and (ii) are true and statement (iii) is not true.

Note: Another way to solve this question is to take some arbitrary value for x, y and z and then verify whether the left hand side is equal to the right hand side or not.