Each of these pairs corresponds to

Each of these pairs corresponds to an edge of the directed graph, with (2, 2) and (3, 3) corresponding
to loops.
▲
The directed graph representing a relation can be used to determine whether the relation
We will study directed
graphs extensively in
Chapter 10.
has various properties. For instance, a relation is reflexive if and only if there is a loop at every
vertex of the directed graph, so that every ordered pair of the form (x, x) occurs in the relation.
A relation is symmetric if and only if for every edge between distinct vertices in its digraph
there is an edge in the opposite direction, so that (y, x) is in the relation whenever (x, y) is
in the relation. Similarly, a relation is antisymmetric if and only if there are never two edges
in opposite directions between distinct vertices. Finally, a relation is transitive if and only if
whenever there is an edge from a vertex x to a vertex y and an edge from a vertex y to a
vertex z, there is an edge from x to z (completing a triangle where each side is a directed edge
with the correct direction).
Remark: Note that a symmetric relation can be represented by an undirected graph, which is a
graph where edges do not have directions.We will study undirected graphs in Chapter 10.
EXAMPLE 10 Determine whether the relations for the directed graphs shown in Figure 6 are reflexive, symmetric,
antisymmetric, and/or transitive.
Solution: Because there are loops at every vertex of the directed graph of R, it is reflexive. R is
neither symmetric nor antisymmetric because there is an edge from a to b but not one from b to
a, but there are edges in both directions connecting b and c. Finally, R is not transitive because
there is an edge from a to b and an edge from b to c, but no edge from a to c.

ผลลัพธ์ (อังกฤษ) 2:

Each of these pairs corresponds to an EDGE of the directed graph, with (2, 2) and (3, 3) corresponding
to loops.
▲
The directed graph representing a relation Can be used to Determine whether the relation
We Will Study directed
graphs extensively. in
Chapter 10.
Various properties has. For instance, a relation is reflexive if and only if there is a loop at Every
Vertex of the directed graph, so that Every ordered Pair of the form (x, x) occurs in the relation.
A relation is symmetric if and only if for. Every EDGE Distinct between vertices in ITS digraph
there is an EDGE in the Opposite direction, so that (Y, x) is in the relation Whenever (x, Y) is
in the relation. Similarly, a relation is antisymmetric if and only if there are Never Two edges
in Opposite directions between Distinct vertices. Finally, a relation is transitive if and only if
Whenever there is an EDGE from a Vertex x to a Vertex Y and an EDGE from a Vertex Y to a
Vertex Z, there is an EDGE from x to Z (completing a Triangle where each Side. EDGE is a directed
with the Correct direction).
Remark: Note that a symmetric relation represented by an undirected graph Can be, which is a
graph where edges do not have Directions.We undirected graphs in Chapter 10. Study Will
Determine whether the EXAMPLE 10. Relations for the directed graphs shown in Figure 6 are reflexive, symmetric,
antisymmetric, and / or transitive.
Solution: Because there are loops at Every Vertex of the directed graph of R, it is reflexive. R is
neither symmetric nor antisymmetric because there is an EDGE from a B to B but not from one to
a, but there are edges in both directions connecting B and C. Finally, R is not transitive because
there is an EDGE from a to B and B to C from an EDGE, but no EDGE from a C to.

ผลลัพธ์ (อังกฤษ) 3:

Each of these pairs corresponds to an edge of the, directed graph with (2 2), and (3 3), corresponding
.

The to loops]. Directed graph representing a relation can be used to determine whether the relation
We will study directed
graphs extensively. In
Chapter 10.
has various properties. For instance a relation, is reflexive if and only if there is a loop at every
vertex. Of the, directed graphSo that every ordered pair of the form (x x), occurs in the relation.
A relation is symmetric if and only if for every. Edge between distinct vertices in its digraph
there is an edge in the, opposite direction so that (y x), is in the relation. Whenever (x y), is
in the relation. Similarly a relation, is antisymmetric if and only if there are never two edges
.In opposite directions between distinct vertices. Finally a relation, is transitive if and only if
whenever there is an. Edge from a vertex x to a vertex y and an edge from a vertex y to a
vertex Z there is, an edge from X to Z (completing a. Triangle where each side is a directed edge
with the correct direction).
Remark: Note that a symmetric relation can be represented. By an, undirected graphWhich is a
graph where edges do not have directions.We will study undirected graphs in Chapter 10.
EXAMPLE 10 Determine. Whether the relations for the directed graphs shown in Figure 6 are reflexive symmetric
antisymmetric and,,, / or transitive.
Solution:? Because there are loops at every vertex of the directed graph of R it is, reflexive. R is
.Neither symmetric nor antisymmetric because there is an edge from a to B but not one from B to
a but there, are edges in. Both directions connecting B and C. Finally R is, not transitive because
there is an edge from a to B and an edge from B. To C but no, edge from a to C.