Two scenerios: (1) A > B and (2) A < B.
(1) Suppose if 1/x > 1/y:
(1/x $ 1/y) = 1/x + 1/y = (x+y)/(xy).
(1/y $ 1/x) = 1/x - 1/y = (y-x)/(xy).
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{1}{x} + \frac{1}{y}) $ (\frac{1}{x} - \frac{1}{y})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x+y}{xy}) $ (\frac{y-x}{xy})\)
Here \((\frac{x+y}{xy})\) must be > \((\frac{y-x}{xy})\). If so, then \((\frac{x+y}{xy}) $ (\frac{y-x}{xy})\) = \((\frac{x+y}{xy}) + (\frac{y-x}{xy})\). Therefore,
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x+y}{xy}) + (\frac{y-x}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x+y+y-x}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{2y}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{2}{x})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = 2A if we suppose 1/x = A.
(1) Suppose if 1/x < 1/y:
(1/x $ 1/y) = 1/y - 1/x = (x-y)/(xy).
(1/y $ 1/x) = 1/y + 1/x = (x+y)/(xy).
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{1}{y} - \frac{1}{x}) $ (\frac{1}{y} + \frac{1}{y})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x-y}{xy}) $ (\frac{x+y}{xy})\)
Here \((\frac{x+y}{xy})\) must be > \((\frac{x-y}{xy})\). If so, then \((\frac{x-y}{xy}) $ (\frac{x+y}{xy})\) = \((\frac{x+y}{xy}) - (\frac{x-y}{xy})\). Therefore,
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x+y}{xy}) - (\frac{x-y}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{x+y-x+y}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{2y}{xy})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = \((\frac{2}{x})\)
\((\frac{1}{x} $ \frac{1}{y}) $ (\frac{1}{y} $ \frac{1}{x})\) = 2A if we suppose 1/x = A.
Therefore, in short, the value of the operation is equal to 2A. A = 1/X in each case and highest A is 1/(1/5) = 5 in D. So 2A in D is 10.
Hope it is clear.