question_answer 70)

Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as a * b Show that Zero is the identity for this operation and each element a fo the set is invertible with 6-a being the inverse of a.  

Answer:

Given a*b =       The composition table for operation * is as follows :      

*	0	1	2	3	4	5
0	0	1	2	.3	4	5
1	1	2	3	4	5	0
2	2	3	4	5	0	1
3	3	4	5	0	1	2
4	4	5	0	1	2	3
5	5	0	1	2	3	4

      From the composition table, it is clear that 0 * 0 = 0, 0 * 1 = 1, 0 * 2 = 2, 0 * 3 = 3  0 * 4 = 4, 0 * 5 = 5.        is an identity element or the operation *. 2n part : Let  be any element of set {0, 1, 2, 3, 4, 5}       Therefore, there exists an element 6 ? a of set {0, 1, 2, 3, 4, 5} such that       a * (6 ? a) = a + 6 ? a ? 6 = 0       and (6 ? a) * a = 6 ? a + a ? 6 = 0       i.e., a * (6 ? a) = (6 ? a) * a = 0        is the inverse of a.       Also 0 * 0 = 0 + 0 = 0        is inverse of itself.