[SOLVED] Abstract Algebra Problem

Hi,

Can anybody help with the following question please:

Let S be the set {a,b,c} and let M(S) be the set of all mappings from S to S.

Consider the subset of P(S) consisting of the mappings x,y,z defined by

x(a) = a, x(b) = b, x(c) = c

y(a) = a, b(b) = c, y(c) = b

z(a) = b, z(b) = b, z(c) = b.

a) Is composition an operation on P(S)? ...I have no because y o z is not defined in P(S) (y o z (a) = c, y o z (b) = c etc)

b)If the answer to a) is no, determine the smallest subset of M(S), Q(S), containing P(S) and such that composition is an operation on Q(S).

I'm not sure how to go about solving part b. Is it necessary to write out all possible operations, their composition and then decide or is there an easier way?

Thanks for any help with this.