Is This Correct Or Not? (Challenge)

$a \circ b = \begin{cases} b+1 \quad , \quad \text{ if } a<b \\ a+2 \quad , \quad \text{ if } a=b\\a+1 \quad , \quad \text{ if } a>b \end{cases}$

If on the left-handed side, the result is either $x+1$ or $y+1$ , then $x\neq y$ . If $x \neq y$ , then the result on the right-handed side shall also be either $x+1$ or $y+1$ , and not $x+2$ . Given both sides satisfy the same condition, the equation shall be TRUE (it shall be equal sign, not $\neq$ . The question said it is $\neq$ , hence the answer is FALSE )

Lets take left hand side:

L.H.S = (x∘y=x+1) OR (x∘y=y+1)

According to rules: x∘y = x+1 if x>y And x∘y = y+1 if y>x We can write,

L.H.S = x>y OR y>x

Either x is greater or y. Thus we can say x≠y

L.H.S = x≠y

Lets take right hand side:

R.H.S = (x∘y≠x+2)

According to first rules: x∘y=x+2=y+2 if x=y (In other words x∘x=x+2)

We can also write this as: x∘y≠x+2≠y+2 if x≠y

R.H.S = x≠y

Left hand side = Right hand side

((x∘y=x+1) OR (x∘y=y+1))=(x∘y≠x+2) Is True

But ((x∘y=x+1) OR (x∘y=y+1))≠(x∘y≠x+2) Is False

Useful Information:

The operator ∘ is called Zeration also known as Hyper0,Increment and Successor.