Theory question 102
Binary OperationsA binary operation * is defined on the set of rational numbers by m * n = (m2 − n2)/(2mn), n ≠ 0 and m ≠ 0. (a) Find −3 * 2. (b) Show whether or not * is associative.
Show the model answer
Model answer
(a) −3 * 2 = (9 − 4)/(2(−3)(2)) = −5/12.
(b) Test with numbers: 1 * 2 = −3/4, so (1 * 2) * 3 = (9/16 − 9)/(2(−3/4)(3)) = 15/8.
2 * 3 = −5/12, so 1 * (2 * 3) = (1 − 25/144)/(2(1)(−5/12)) = −119/120.
15/8 ≠ −119/120, so * is not associative.
Explanation
One counter-example is enough to show an operation is not associative.