A binary operation * is defined on the set of real numbers R, by p*q = p + q - \(\frac{pq}{2}\), where p, q \(\in\) R. Find the: (a) inverse of -1 under * g...
Question 1 Report
A binary operation * is defined on the set of real numbers R, by p*q = p + q - \(\frac{pq}{2}\), where p, q \(\in\) R. Find the:
(a) inverse of -1 under * given that the identity clement is zero.
(b) truth set of m* 7 = m* 5,
Answer Details
The operation is \( p * q = p + q - \dfrac{pq}{2} \), with identity element \(0\).
(a) Inverse of \(-1\). Let \(x\) be the inverse of \(-1\), so \( (-1) * x = 0 \):
\[ -1 + x - \frac{(-1)x}{2} = 0 \Rightarrow -1 + x + \frac{x}{2} = 0 \Rightarrow \frac{3x}{2} = 1 \Rightarrow x = \frac{2}{3}. \]
The inverse of \(-1\) is \( \dfrac{2}{3} \).
(b) Truth set of \( m * 7 = m * 5 \).
\( m * 7 = m + 7 - \dfrac{7m}{2} \) and \( m * 5 = m + 5 - \dfrac{5m}{2} \). Setting them equal:
\[ 7 - \frac{7m}{2} = 5 - \frac{5m}{2} \Rightarrow 7 - 5 = \frac{7m}{2} - \frac{5m}{2} \Rightarrow 2 = m. \]
The truth set is \( \{2\} \).
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