A binary operation + defined on the set R of real numbers by a+b= a+b +ab where a bAR find the operation +the identity element
To find the identity element for the binary operation + defined on the set R of real numbers, we need to find a number, let's call it "e," such that for any real number "a," the equation a + e = e + a = a holds.
Let's proceed with finding the value of e by substituting the definition of the operation + into the equation:
a + e = a + e + ae
Now, we can see that the term "ae" will play a role in obtaining the identity element. To isolate this term, we subtract both sides of the equation by "a":
e = e + ae
Next, we subtract "e" from both sides of the equation:
0 = ae
Now, since we want to find a value for "e" that works for any real number "a," we need to consider two cases:
1. If a = 0:
We substitute a = 0 into the equation ae = 0, which gives us:
0 = 0e
This means that for any value of e, the equation will still hold when a = 0.
2. If a ≠ 0:
We divide both sides of the equation ae = 0 by a:
e = 0
By examining both cases, we can conclude that the identity element for the defined operation + on the set R of real numbers is 0.