Say whether each of the following collections of subsets is a partition of the g

Question

Say whether each of the following collections of subsets is a partition of the given set. If not, explain why not. (a) As subsets of the provincial ridings of New Brunswick, the set of ridings where a PC candidate won, the set where a Liberal candidate won, and the set where an NDP candidate won. (The question refers to the election taking place on Monday; the correct answer will depend on the outcome of the election.) (b) As subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9}, the sets {5, 4, 1}, {3, 2, 6}, {9}, {7, 8}. (c) As subsets of the months, those whose name contains an y, those with six or fewer letters in its name, those whose name ends in r. (d) As subsets of the US states, the states that begin with A, those that begin with B, those that begin with C, and so on (26 sets in total).

Explanation / Answer

(a) I don't know what election they are talking about, but if each riding only corresponds to one party, if each party of the three aforementioned parties won at least one riding, and no other parties won a riding, then it should be a partition. Otherwise not.

(b) This is a partition. The subsets are nonempty, do not intersect, and their union is the whole set {1, ..., 9}

(c) This is not a partition. Note that "July" is both in the "six or fewer letters" set and the "name contains a y" set, so this collection of subsets fails the "mutually exclusive" part of the definition of a partition.

(d) This is not a partition because some of the subsets are empty. For instance, there are no states that begin with the letter Z.

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