A binary operation x defined on the set of integers is such that m x n =m+n+mn for all the integers m and n. Find the inverse of -5 under the operation, if the Identity element is o?
we want n = m^-1, so that means -5 x n = 0
-5 x n = -5 + n - 5n = -5 - 4n
n = -5/4
am I missing something? n should be an integer
maybe the set of integers is not closed under "x"
Wow. Very correct
To find the inverse of -5 under the binary operation x, we need to find a number that, when combined with -5 using this operation, gives the identity element, which is 0.
Let's denote the inverse of -5 as y, so we have -5 x y = 0.
Using the given binary operation definition, substitute the values -5 and y into the equation:
(-5) x y = (-5) + y + (-5)y = 0
Now, we can solve this equation to find the value of y.
Rearranging the equation, we get:
-5 + y + (-5)y = 0
Combine like terms:
(-5)y + y - 5 = 0
Factor out y:
y((-5) + 1) - 5 = 0
Simplify:
-4y - 5 = 0
Add 5 to both sides:
-4y = 5
Divide both sides by -4:
y = 5 / (-4)
y = -5/4
Therefore, the inverse of -5 under the operation x is -5/4.