a *b=a+b-2ab
Find the identity element,inverse element
To find the identity element and inverse element in an algebraic structure, we need to determine the values that satisfy the given equation:
a * b = a + b - 2ab
1. Identity element (e):
The identity element (e) is a value that, when combined with any other value in the algebraic structure, leaves the operation unchanged. So, for any value 'a,' we assume that there is an identity element 'e' such that a * e = a.
Substituting 'e' into the given equation, we have:
a * e = a + e - 2ae
To maintain the equality, we can equate the corresponding terms:
a = a + e - 2ae
By comparing the coefficients, we get two equations:
(1) a = a
(2) 0 = e - 2ae
From equation (1), we see that a cancels out, leaving us with 0 = e - 2ae.
To find the identity element, we solve equation (2) for 'e':
0 = e - 2ae
2ae = e
e(2a - 1) = 0
From this expression, we have two possibilities:
1) e = 0 when (2a - 1) ≠ 0
2) No value of e exists when (2a - 1) = 0
Therefore, the identity element is e = 0.
2. Inverse element (a'):
The inverse element of a given value 'a' is a value 'a' such that a * a' = e, where 'e' is the identity element.
Substituting 'a' and 'a'' into the given equation, we have:
a * a' = a + a' - 2aa'
To find the inverse element, we need to determine the value of 'a' such that a + a' - 2aa' = e = 0.
Using the equation 0 = a + a' - 2aa', we can equate the corresponding terms:
1 = a + a' - 2aa'
Rearrange the equation and isolate 'a', we get:
2aa' - a' + a - 1 = 0
Consider this equation as a quadratic equation in 'a', and solve for 'a':
2aa' - a' + a - 1 = 0
By comparing the coefficients, we have two equations:
(1) 2aa' + a - 1 = 0
(2) -a' = 0
From equation (2), we see that a' = 0.
Substituting a' = 0 into equation (1), we have:
2a(0) + a - 1 = 0
a - 1 = 0
a = 1
Therefore, the inverse element of 'a' is a' = 0, and this is true for any value 'a'.
To summarize:
- The identity element is e = 0.
- The inverse element of 'a' is a' = 0 for any value 'a'.