tpa, bq a b is a multiple of 5 u tp0, 0q, p0, 5q, p0, 5q,...,

Transcription

1 A binar relation on a set A is a sbset of A ˆ A, hereelements pa, bq are ritten as a b. For eample, let A Z, so A ˆ A tpn, mq n, m P Z. Let be the binar relation gien b a b if and onl if a and b hae the same remainder modlo 5. In terms of sbsets, the binar relation is defined as the sbset of Z ˆ Z gien b tpa, bq a b is a mltiple of 5 tp0, 0q, p0, 5q, p0, 5q,...,...,p1, 1q, p1, 6q, p1, 4q,...,...,p 1, 1q, p 1, 4q, p 1, 6q,... A binar relations models relationships beteen elements of A. More eamples of (binar) relations: 1. For A anmbersstem,leta b if a b. R, S, T 2. For A anmbersstem,leta b if a b. R, not S, T 3. For A anmbersstem,leta b if a b. not R, not S, T 4. For A a set of people, let a b if a knos b. 5. For A a set of people, let a b if a and b speak a common langage. R, S, not T A binar relation on a set A is... (R) refleie if a a for all a P A; (S) smmetric if heneer a b, thenb a as ell; (T) transitie if heneer a b and b c, thena c as ell. An eqialence relation on a set A is a binar relation that is refleie, smmetric, and transitie. (Onl #1)

2 Define the relation on Z gien b a b if a and b hae the same remainder hen diided b 5. Is is an eqialence relation? Check: To integers a and b hae the same remainder if and onl if a b is a mltiple of 5. So a b if and onl if a b 5n for some n P Z. refleiit: smmetr: transitiit: a a 0 (a has the same remainder as itself) X If a b 5n, thenb a 5n 5p nq. X If a b 5n and b c 5m, then a c pa bq`pb cq 5n ` 5m 5pn ` mq.x Yes! This is an eqialence relation! Let A be a set. Consider the relation on PpAq b Is is an eqialence relation? S T if S Ä T. Check: This is transitie, bt not refleie or smmetric. So no, it is not an eqialence relation. Is S T if S Ñ T an eqialence relation on PpAq? Check: This is refleie and transitie, bt not smmetric. So no, it is not an eqialence relation. Is S T if S Ñ T or S Ñ T an eqialence relation on PpAq? Check: This is refleie and smmetric, bt not transitie. So still no, it is not an eqialence relation.

3 Let A ta, b, c. WhichsbsetsS of A ˆ A define eqialence relations on A? Anser Refleie: S mst contain pa, aq, pb, bq, and pc, cq (sometimes ritten a a, b b, andc c in this contet). Smmetric: If P S, thenemstalsohae (for an, Pta, b, c). Transitie: If P S and z P S, thenemstalsohae z (for an,, z Pta, b, c). Or approach: Start ith a a, b b, c c, andthenadd elements one at a time, and see hat else is forced to be in S ta a, b b, c c 2. ta a, b b, c c, a b, b a 3. ta a, b b, c c, a c, c a 4. ta a, b b, c c, b c, c b 5. ta a, b b, c c, a b,a c, b a, c a, c b, b c That s it! Let be an eqialence relation on a set A, andleta P A. The set of all elements b P A sch that a b is called the eqialence class of a, denoted b ras. Eample: We shoed that for A ta, b, c the set S ta a, b b, c c, a c, c a defines an eqialence relation on A. Then ras ta, c rcs, and rbs tb are the to eqialence classes in A (ith respect to this relation). (We sa there are to, not three, since the eqialence classes refers to the sets themseles, not to the elements that generate them.)

4 Let be an eqialence relation on a set A, andleta P A. The set of all elements b P A sch that a b is called the eqialence class of a, denoted b ras. Eample: We shoed that a b if a and b hae the same remainder hen diided b 5. is an eqialence relation on Z. Then r0s t5n n P Z r1s t5n`1 n P Z r2s t5n`2 n P Z r3s t5n ` 3 n P Z r4s t5n ` 4 n P Z r5s t5n ` 5 n P Z t5m m P Z r0s r 5s r10s r6s t5n ` 6 n P Z t5m`1 m P Z r1s r 4s r11s. In general, if Prs, thatmeans. So. So Prs. Using this reasoning, e can sho that if Prs, thenrs rs. We call an element a of a class C representatie of C (since e can rite C ras for an a P C). Let be an eqialence relation on a set A, andleta P A. The set of all elements b P A sch that a b is called the eqialence class of a, denoted b ras. In fact, the eqialence classes of A partition A into sbsets. This means that 1. the eqialence classes are sbsets of A: ras Ñ A for all a P A; 2. an to eqialence classes are either eqal or disjoint: for all a, b P A, eitherras rbs or rasxrbs H; and 3. the nion of all the eqialence classes is all of A: A aparas. For eample, in or last eample, there are eactl 5 eqialence classes: r0s, r1s, r2s, r3s, and r4s. An other seemingl di erent class is actall one of these (for eample, r5s r0s). And r0syr1syr2syr3syr4s Z.

5 A graph is a set of objects, or ertices, together ith a (mlti)set of edges that connect pairs of ertices. Eample: e 2 e 3 e 4 e 5 Here, the ertices are V t,,,,, and the edges are E t, e 2, e 3, e 4, e 5. An edge that connects a erte to itself (like e 5 )iscalledaloop. We sa a erte a is adjacent to a erte b if there is an edge connecting a and b. (Notice that for a generic graph, adjacenc is a smmetric relation, bt is not refleie nor is it transitie.) Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. Simple! NOT simple! NOT simple!

6 Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. A graph is a mltigraph if there are no loops, bt there cold be mltiple edges beteen to ertices. Mltigraph! Mltigraph! NOT Mltigraph! Classes of graphs: A graph is simple if there are no loops and eer pair of ertices has at most one edge beteen them. A graph is a mltigraph if there are no loops, bt there cold be mltiple edges beteen to ertices. A graph is a psedograph if there cold be loops or mltiple edges. (This is jst hat e call a graph.) So t psedographs/graphs â t mltigraphs â t simple graphs. (Note: The â smbol is sed here becase, for eample, eer simple graph is a mltigraph, bt there are mltigraphs that are not simple.)

7 Directed graphs A directed graph (also called a digraph or a qier) is a graph, together ith a choice of direction for each edge. A directed graph is simple if there are no loops and eer pair of ertices has at most one edge in each direction beteen them. Simple! NOT Simple! NOT Simple! Directed graphs A directed graph (also called a digraph or a qier) is a graph, together ith a choice of direction for each edge. A directed graph is simple if there are no loops and eer pair of ertices has at most one edge in each direction beteen them. A directed graph is a directed mltigraph if there cold be loops or mltiple edges. (This is jst hat e call a directed graph) So t directed (mlti)graphs â t directed simple graphs. The book also talks abot mied graphs, here some of the edges are directed and some aren t. We sall take care of this b modeling the non-directed edges ith to directed edges, one in each direction.

8 G e 2 e 3 e 4 e 5 Npq t, Npq t Npq t Npq t Npq t, Npt, q t,, Npt, q t,,, We sa a erte a is adjacent to a erte b if there is an edge connecting a and b. For eample, in G, is adjacent to and ; is adjacent to ; is adjacent to and. We sa that an edge is incident to a erte if the edge connects to the erte. For eample, in G, is incident to and ; e 5 is incident to. If to ertices and are adjacent, e e sa that the are neighbors, andthat is in the neighborhood Npq of (and ice-ersa). If A Ñ V,then NpAq PA Npq. G e 2 e 3 e 4 e 5 Npq t, Npq t Npq t Npq t Npq t, degpq 3 degpq 1 degpq 2 degpq 1 degpq 3 The degree degpq of a erte is the nmber of edge ends attached to. Fact: a graph is simple if and onl if degpq Npq for all P V. H e 2 e 3 Npq t, Npq t Npq t Npq t Npq t degpq 2 degpq 1 degpq 1 degpq 1 degpq 1

9 G e 2 e 3 e 4 e 5 Npq t, Npq t Npq t Npq t Npq t, degpq 3 degpq 1 degpq 2 degpq 1 degpq 3 The degree degpq of a erte is the nmber of edge ends attached to. We call a graph reglar if all the ertices hae the same degree. Theorem (The handshake theorem) In a graph G pv,eq, 2 E ÿ PV deg. Corollar In an graph, there are an een nmber of odd ertices.