The operation * is defined on the set r of rea- numbers by a*b=-1 , for all a, b, £ r is r closed under * is the operation *communicative in r is the operation * association in r.
The operation * is commutative in r if for every pair of real numbers a and b, a * b = b * a.
To check if * is commutative in r, we need to verify if -1 = -1 holds true for all real numbers a and b.
For example, let's take a = 2 and b = 3.
a * b = 2 * 3 = -1
b * a = 3 * 2 = -1
Since a * b = b * a = -1, the operation * is commutative in r.
The operation * is associative if for every triple of real numbers a, b, and c, the expression (a * b) * c is equal to a * (b * c).
To check if * is associative in r, we need to verify if (a * b) * c = a * (b * c) holds true for all real numbers a, b, and c.
Let's take a = 1, b = 2, and c = 3.
(a * b) * c = (1 * 2) * 3 = (-1) * 3 = -3
a * (b * c) = 1 * (2 * 3) = 1 * (-1) = -1
Since (a * b) * c = -3 and a * (b * c) = -1, the operation * is not associative in r.