Combinatorics Through Guided Discovery, 2004a

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A.2. Equivalence relations 139 ⇒ Problem 345. Now suppose again we are choosing three distinct flavors of ice cream out of four, but instead of putting scoops in a cone or choosing pints, we are going to have the three scoops arranged symmetrically in a circular dish. Similarly to choosing three pints, we can describe a selection of ice cream in terms of which one goes in the dish first, which one goes in second (say to the right of the first), and which one goes in third (say to the right of the second scoop, which makes it to the left of the first scoop). But again, two of these lists will sometimes be equivalent. Once they are in the dish, we can’t tell which one went in first. However, there is a subtle difference between putting each flavor in its own small dish and putting all three flavors in a circle in a larger dish. Think about what makes the lists of flavors equivalent, and draw the directed graph whose vertices consist of all lists of three of the flavors of ice cream and whose edges connect two lists that we cannot tell the difference between as dishes of ice cream. How many dishes of ice cream can we distinguish from one another? (h) Problem 346. Draw the digraph for Problem 38 in the special case where we have four people sitting around the table. In Problems 344, 345, and 346 (as well as Problems 34, 38, and 39) we can begin with a set of lists, and say when two lists are equivalent as representations of the objects we are trying to count. In particular, in Problems 344, 345, and 346 you drew the directed graph for this relation of equivalence. Technically, you should have had an arrow from each vertex (list) to itself. This is what we mean when we say a relation is reflexive. Whenever you had an arrow from one vertex to a second, you had an arrow back to the first. This is what we mean when we say a relation is symmetric. When people sit around a round table, each list is equivalent to itself: if List1 and List 2 are identical, then everyone has the same person to the right in both lists (including the first person in the list being to the right of the last person). To see the symmetric property of the equivalence of seating arrangements, if List1 and List2 are different, but everyone has the same person to the right when they sit according to List2 as when they sit according to List1, then everybody better have the same person to the right when they sit according to List1 as when they sit according to List2. In Problems 344, 345 and 346 there is another property of those relations you may have noticed from the directed graph. Whenever you had an arrow from L 1 to L 2 and an arrow from L 2 to L 3 , then there was an arrow from L 1 to L 3 . This is what we mean when we say a relation is transitive. You also undoubtedly noticed how the directed graph divides up into clumps of mutually connected vertices. This is what equivalence relations are all about. Let’s be a bit more precise in our description of what it means for a relation to be reflexive, symmetric or transitive. • If R is a relation on a set X,wesayR is reflexive if (x, x) ∈ R for every x ∈ X.
140 A. Relations • If R is a relation on a set X, wesayR is symmetric if (x, y) is in R whenever (y, x) is in R. • If R is a relation on a set X, wesayR is transitive if whenever (x, y) is in R and (y, z) is in R, then (x, z) is in R as well. Each of the relations of equivalence you worked with in Problems 344, 345 and 346 had these three properties. Can you visualize the same three properties in the relations of equivalence that you would use in Problems 34, 38, and 39? We call a relation an equivalence relation if it is reflexive, symmetric and transitive. After some more examples, we will see how to show that equivalence relations have the kind of clumping property you saw in the directed graphs. In our first example, using the notation (a, b) ∈ R to say that a is related to B is going to get in the way. It is really more common to write aRb to mean that a is related to b. For example, if our relation is the less than relation on {1, 2, 3}, you are much more likely to use x < y than you are (x, y) ∈
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