To find: identity and inverse element

For a binary operation if a*e = a, then e s called the right identity

If e*a = a then e is called the left identity

For the given binary operation,

e*b = b

⇒ e + b = b

⇒ e = 0 which is less than 6.

b*e = b

⇒ b + e = b

⇒ e = 0 which is less than 6

For the 2^nd condition,

e*b = b

⇒ e + b - 6 = b

⇒ e = 6

But e = 6 does not belong to the given set (0,1,2,3,4,5)

So the identity element is 0

An element c is said to be the inverse of a, if a*c = e where e is the identity element (in our case it is 0)

a*c = e

⇒ a + c = e

⇒ a + c = 0

⇒ c = - a

a belongs to (0,1,2,3,4,5)

- a belongs to (0, - 1, - 2, - 3, - 4, - 5)

So c belongs to (0, - 1, - 2, - 3, - 4, - 5)

So c = - a is not the inverse for all elements a

Putting in the 2^nd condition

a*c = e

⇒ a + c - 6 = 0

⇒ c = 6 - a

0≤a<6

⇒ - 6≤ - a<0
⇒ 0≤6 - a<6
0≤c<5

So c belongs to the given set

Hence the inverse of the element a is (6 - a)

Hence proved