DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

Chapter 3
Equivalence Relations
3.1 Equivalence Relations
Equivalence relations have been in the background of some of the problems
you’ve already worked. For instance, in Problem 9 with three distinct flavors
at first it was probably tempting to say there are 12 flavors for the first pint,
11 for the second, and 10 for the third, so there are 12 · 11 · 10 ways to
choose the pints of ice cream. However, once the pints have been chosen,
bought, and put into a bag, there is no way to tell which one was bought
first, which second, and which third. The number 12 · 11 · 10 is the number
of lists of three distinct flavors, in which the order in which the pints are
bought makes a difference. Two of the lists become equivalent after the ice
cream purchase if they have the same flavors of ice cream. In other words,
two of these lists are equivalent (are related for this problem) if they list the
same subset of the set of twelve ice cream flavors. To visualize this relation
with a digraph, one vertex would be needed for each of the 12·11·10 = 1320
lists (which is not feasible to draw). Even with five flavors of ice cream the
number of vertices would be 5 · 4 · 3 = 60. So for now the easier-to-draw
question of choosing three pints of different ice cream flavors from a choice
of four flavors of ice cream will be considered. For this, there are 4·3·2 = 24
different lists.
◦88. Suppose you have four flavors of ice cream: V(anilla), C(hocolate),
S(trawberry) and P(each). Draw the directed graph whose vertices
are all lists of three distinct flavors of the ice cream, and whose edges
connect two lists if they list the same three flavors. This graph makes
it pretty clear in how many “really different” ways you may choose 3
flavors from four. How many?
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