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...

Answer Details

(a) To find the inverse of -1 under *, we need to find a real number x such that -1 * x = x * (-1) = 0, where 0 is the identity element under *.

Substituting -1 and x into the given formula, we get:

-1 * x = -1 + x - \(\frac{(-1)x}{2}\) = 0

Simplifying this equation, we get:

x - \(\frac{x}{2}\) = 1

\(\frac{x}{2}\) = 1

x = 2

Therefore, the inverse of -1 under * is 2.

(b) To find the truth set of m * 7 = m * 5, we need to find all real numbers m that satisfy the equation.

Substituting m and 7 into the given formula, we get:

m * 7 = m + 7 - \(\frac{7m}{2}\)

Substituting m and 5 into the given formula, we get:

m * 5 = m + 5 - \(\frac{5m}{2}\)

Setting these two expressions equal to each other, we get:

m + 7 - \(\frac{7m}{2}\) = m + 5 - \(\frac{5m}{2}\)

Simplifying this equation, we get:

\(\frac{m}{2}\) = 2

m = 4

Therefore, the truth set of m * 7 = m * 5 is {4}, which means that the only real number that satisfies the equation is 4.