Theory question 101
Binary OperationsA binary operation * is defined on the set of real numbers, R, by p * q = p + q − pq/2, where p, q ∈ R. Find the: (a) inverse of −1 under * given that the identity element is zero. (b) truth set of m * 7 = m * 5.
Show the model answer
Model answer
(a) p * p⁻¹ = 0: p + p⁻¹ − pp⁻¹/2 = 0, so p⁻¹(1 − p/2) = −p and p⁻¹ = −2p/(2 − p).
For p = −1: p⁻¹ = 2/3 (check: −1 + 2/3 + 1/3 = 0).
(b) m + 7 − 7m/2 = m + 5 − 5m/2, so 2 = m. Truth set {2}.
Explanation
Check the inverse by substituting back.