x & y is defined as |x-y|. Which of following is (are) t

\(x\) & \(y\) is defined as \(|x−y|\)

I. \(x\) & \(y = y\) & \(x\)
\(\Rightarrow |x - y| = |y - x|\)
\(\Rightarrow |x - y| = |x - y|\) (Since \(|2| = |-2|\quad{\color{green}\checkmark})\) - TRUE

II. \(2x\) & \(2y\) = \(2(x\)&\(y)\)
\(Rightarrow |2x - 2y| = 2|x - y|\)
\(\Rightarrow |2(x - y)| = 2|x - y|\)
\(\Rightarrow 2|x - y| = 2|x - y|\quad{\color{green}\checkmark}\) - TRUE

III. \(x\) & \(x = 0\)
\(\Rightarrow |x - x| = |0| = 0 \quad{\color{green}\checkmark}\) - TRUE

=>

Statement I.
x & y = | x – y | = | y – x | = y & x.
Statement I is true.

Statement II
2x & 2y = | 2x – 2y | = 2| x – y | = 2 (x&y).
Statement II is true.

Statement III
x & x = | x – x | = 0.
Statement III is true.

Therefore, E is the answer.
Answer: E