Question

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. Write the inverse of 4.

Answer 1

The given binary operation is a*b = a+b+2 

In order to find the inverse of the relation, we have to find the identity element first.

Let that identity element be e then 

a*e = a 

From que. 

a*e = a+e+2 

So, from the above two relations we have 

a+e+2 = a 

or, e+2 = 0 

∴ e = -2 

Hence the identity element is -2 for this binary operation. 

Now let a’ be the inverse of this relation 

Then as per the definition of the inverse element 

a*a’ = e 

∴ a+a’+2 = -2 

∴ a’ = -4 -a 

And for 4 ,i.e. a = 4 

a’ = -4 – 4 

∴ a’ = -8 

Thus the inverse element of 4 is -8 for the given binary operation.