Since {1, 2, . . . , 6} is the set of all possible outcomes of athrow with a regular die, the set of all possible outcomes of athrow with two dice is Throws := {1, 2, . . . , 6} × {1, 2, . . . ,6}. We define eleven subsets P2, P3, . . . , P12 of Throws asfollows: Pk := {: m + n = k} for k ∈ {2, 3, . . . ,12}. For example, P3 is the set of all outcomes for which the sumof the two numbers of dots thrown is 3.
(a) Show that the sets P2, P3, . . . , P12 form a partition ofthe set Throws.
(b) Let R be the equivalence relation on Throws that has P2, P3,. . . , P12 as its equivalence classes. Give a definition of R bymeans of a description.
(c) Give a complete system of representatives for theequivalence relation R.
