Discrete Structure

Discrete Structure Li Tak Sing(李德成)

Equivalence relations • Any binary relation that is reflexive, symmetric, and transitive is called an equivalence relation.

Sample equivalence relations • R is over N, xRy means x+y is even • R is over real number, xRy means sin(x)=sin(y)

Examples • Let R be defined on N by xRy iff |x-y| is odd. Show that R is not an equivalance relation on N. • Given the relation over the integers defined by a~b iff |a|=|b|, either prove that ~ is an equivalence relation or prove that ~ not an equivalence relation.

Intersection property of equivalence • If E and F are equivalence relations on the set A, then EF is an equivalence relation on A.

Kernel relations • If f is a function with domain A, then the relation ~ defined by x~y iff f(x)=f(y)is an equivalence relation on A, and it is called the kernel relation of f.

Equivalence classes • Let R be an equivalence relation on a set S. If aS, then the equivalence class of a, denoted by [a], is the subset of S consisting of all elements that are equivalent to a. In other words, we have [a]={x S | xRa}

Example of equivalence class • a~b iff a+b is even • a~b iff sin(a)=sin(b) • a~b iff a and b are studying the same course

Property of equivalences • Let S be a set with an equivalence relation R. If a, b S, then either [a]=[b] or [a][b]=. • Proof. If [a][b], and [a][b]  , there is an element c so that c[a] and c[b]. So we have aRc and cRb, therefore aRb. Now, for any x [a] , xRa. Since aRb, therefore we have xRb which implies that x [b]. Therefore, [a][b]. Similarly, we have [b][a].

Partitions • By a partition of a set we mean a collection of nonempty subsets that are disjoint from each other and whose union is the whole set. • If R is an equivalence relation on the set S, then the equivalence classes form a partition of S. Conversely, if P is a partition of a set S, then there is an equivalence relation on S whose equivalence classes are sets of P.

Refinement of a partition • Suppose that P and Q are two partitions of a set S. If each set of P is a subset of a set in Q, then P is a refinement of Q. The finest of all partitions on S is the collection of singleton sets. The coarsest of all partitions of S is the set S itself.

Example • aRb iff ab (mod 2) • aSb iff ab (mod 4)

Example • Given the following set of words. {rot, tot, root, toot, roto, toto, too, to otto}. • Let f be the funtion that maps word to its set of letters. For kernel relation of f, describe the equivalence classes. • Let f be the function that maps a word to its bag of letters. For the kernel relation of f, describe the equivalence classes.

Kruskai's algorithm for minimal spanning trees • In the spanning tree problems, we can define a relation R so that aRb if there is a path between a and b. The algorithm is:

Kruskal's Algorithm • Sort the edges of the graph by weight, and let L be the sorted list. • Let T be the minimal spanning tree and initialize T:=. • For each vertex v of the graph, create the equivalence class [v]={v} • while there are 2 or more equivalence classes do Let {a,b} be the edge at the head of L; L:=tail(L); if [a][b] then T:=T{{a,b}}; Replace the equivalence classes [a] and [b] by [a] [b] fi • od

Spanning trees

Discrete Structure