Is This Correct Or Not? (Challenge)

Logic Level 2

We define the binary operator, \( \circ\) such that \( a \circ a = a+2\), \(a \circ b = \begin{cases} a+1 \quad , \quad \text{ if } a>b \\ b+1 \quad , \quad \text{ if } b>a \end{cases} \).

Is this statement correct?

\[ (x\circ y = x+1) \text{ OR } (x\circ y = y+1) \ne (x\circ y \ne x+2) \]

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