A binary operation * is defined on set r of real number by : a*b=a+b+ab where a,b er (i) calculate 5*(-2)*5 (ii) find the identity element e of R under the * (iii) determine the inverse under * of a general element a ER and state which element has no inverse
(i) To calculate 5*(-2)*5, we substitute a = 5 and b = -2 into the expression for the binary operation:
5*(-2)*5 = 5 + (-2) + 5*(-2) = 5 - 2 - 10 = -7.
(ii) To find the identity element e of R under the * operation, we need to determine an element e such that a*e = e*a = a for all a in R. So, let's consider a general element a:
a*e = a + e + a*e.
Since we want this to equal a for any value of a, we can equate the coefficients of e on both sides:
1 = 1 + e.
From this equation, we can see that e must be 0. We can confirm this by substituting e = 0 in the expression for the binary operation:
a*0 = a + 0 + a*0 = a.
So, the identity element e of R under the * operation is 0.
(iii) To determine the inverse under * of a general element a ER, we need to find an element x such that a*x = x*a = e. Let's consider the general expression for the binary operation:
a*x = a + x + a*x.
To simplify this equation and solve for x, we can rearrange terms:
a*x - a*x = a + x - a.
The left side cancels out:
0 = x.
Therefore, the inverse of any element a in R under the * operation is x = 0.
Since every element in R has an inverse (the identity element 0), there is no element that does not have an inverse under the * operation.