A binary operation * is defined on the set of real numbers R, by p*q...
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,
Explanation
p*e = e*p = p
p + e - \(\frac{pe}{2}\) = p
e - \(\frac{pe}{2}\) = 0
e(\(1 - \frac{p}{2}\)) = 0
e = 0
p*p\(^{-1}\) = e
p - p\(^{-1}\) - \(\frac{pp^{-1}}{2}\) = 0
p\(^{-1}\) - \(\frac{pp}{2}\) = -p
p\(^{-1}\) (1 - \(\frac{p}{2}\)) = -p
p\(^{-1}\) = \(\frac{-p}{1 - \frac{p}{2}}\)
Inverse of -1 \(\to\) p = -1
p\(^{-1}\) = \(\frac{-1}{1 - (\frac{-1}{2})} = \frac{-1}{\frac{3}{2}}\)
= \(\frac{-2}{3}\)
(b) m*7 = m + 7 - \(\frac{7m}{2}\), m*5 = m + 5 - \(\frac{5m}{2}\)
m*7 = m*5, m + 7 - \(\frac{7m}{2}\) = m + 5 - \(\frac{5m}{2}\)
7 - 5 = -\(\frac{5m}{2} + \frac{7m}{2}\)
2 = \(\frac{2m}{2}\)
m = 2