An operation * is defined on the set of real numbers,R, by x *=x+y+3xy if * is commutative (a) find the Identity element of R under the operation (b) determine the inverse of x ER for which the value exist
(a) To find the identity element of R under the operation *, we need to find a real number e such that for any real number x, x * e = x and e * x = x.
Let's consider x * e. Using the definition of the operation *, we have x * e = x + e + 3xe. Since we want this to be equal to x for any x, we can equate the coefficients of x on both sides of the equation. We get 1 + 3e = 1, which means e = 0.
Now let's consider e * x. Using the definition of the operation *, we have e * x = e + x + 3ex. Since we want this to be equal to x for any x, we can equate the coefficients of x on both sides of the equation. We get 1 + 3e = 1, which again means e = 0.
Therefore, the identity element of R under the operation * is 0.
(b) To determine the inverse of x in R for which the value exists, we need to find a real number y such that x * y = 0 and y * x = 0.
Using the definition of the operation *, we have x * y = x + y + 3xy. Setting this equal to 0, we get x + y + 3xy = 0. Rearranging the equation, we have y = -(x + 3xy).
Now let's consider y * x. Using the definition of the operation *, we have y * x = y + x + 3yx. Setting this equal to 0, we get y + x + 3yx = 0. Rearranging the equation, we have y = -(x + 3yx).
From the above equations, we can see that y = -(x + 3xy) = -(x + 3yx). Therefore, the inverse of x in R exists if and only if -(x + 3xy) = -(x + 3yx).
Note that if x = 0, then the equation -(x + 3xy) = -(x + 3yx) is satisfied for any y. This means that the inverse of x = 0 is any real number y.
If x is not equal to 0, then we can simplify the equation by dividing both sides by -(x + 3yx). We get 1 = 1. This equation is always true regardless of the value of x or y. Therefore, the inverse of any non-zero x in R does not exist.