Theory question 101
Binary OperationsA binary operation * is defined on the set of real numbers, R, by x * y = 2x + 2y − xy/5. Find the: (a) inverse of x under the operation *.
(b) truth set when m * 7 = −2 * m.
Show the model answer
Model answer
(a) Identity: x * e = x gives 2x + 2e − xe/5 = x, so e(10 − x)/5 = −x and e = −5x/(10 − x).
(Note: e depends on x, so strictly the operation has no single identity; following the question:)
x * x⁻¹ = e: 2x + x⁻¹(2 − x/5) = −5x/(10 − x), so x⁻¹(10 − x)/5 = (2x² − 25x)/(10 − x).
x⁻¹ = 5(2x² − 25x)/(10 − x)², x ≠ 10.
(b) m * 7 = 2m + 14 − 7m/5; −2 * m = −4 + 2m + 2m/5.
14 − 7m/5 = −4 + 2m/5, so 18 = 9m/5 and m = 10. Truth set {10}.
Explanation
Find the identity element first, then use x * x⁻¹ = e.