0.6 Discrete structures relation  (Page 6/13)

Definition (transitive relation): A relation R on a set A is called transitive if and only if for any a, b, and c in A, whenever<a, b>∈R , and<b, c>∈R ,<a, c>∈R .

Example 8: The relation ≤ on the set of integers {1, 2, 3} is transitive, because for<1, 2>and<2, 3>in ≤,<1, 3>is also in ≤, for<1, 1>and<1, 2>in ≤,<1, 2>is also in ≤, and similarly for the others. As a matter of fact ≤ on any set of numbers is also transitive. Similarly ≥ and = on any set of numbers are transitive.

Figure 5 show the digraph of relations with different properties.

(a) is reflexive, antisymmetric, symmetric and transitive, but not irreflexive.

(b) is neither reflexive nor irreflexive, and it is antisymmetric, symmetric and transitive.

(c) is irreflexive but has none of the other four properties.

(d) is irreflexive, and symmetric, but none of the other three.

(e) is irreflexive, antisymmetric and transitive but neither reflexive nor symmetric.

Operations on binary relations

Set operations

A relation is a set. It is a set of ordered pairs if it is a binary relation, and it is a set of ordered n-tuples if it is an n-ary relation. Thus all the set operations apply to relations such as ∪, ∩, and complementing.

For example, the union of the "less than" and "equality" relations on the set of integers is the "less than or equal to" relation on the set of integers. The intersection of the "less than" and "less than or equal to" relations on the set of integers is the "less than" relation on the same set. The complement of the "less than" relation on the set of integers is the "greater than or equal to" relation on the same set.

Composite relations

If the elements of a set A are related to those of a set B, and those of B are in turn related to the elements of a set C, then one can expect a relation between A and C. For example, if Tom is my father (parent-child relation) and Sarah is a sister of Tom (sister relation), then Sarah is my aunt (aunt-nephew/niece relation). Composite relations give that kind of relations.

Definition(composite relation): Let R1 be a binary relation from a set A to a set B, R2 a binary relation from B to a set C. Then the composite relation from A to C denoted by R1R2 (also denoted by R1 ∘ R2 is defined as

R1R2 = {<a, c>| a ∈A ⋀c ∈C ⋀∃b [b ∈B ⋀<a, b>∈R1 ⋀<b, c>∈R2 ] } .

In English, this means that an element a in A is related to an element c in C if there is an element b in B such that a is related to b by R1 and b is related to c by R2. Thus R1R2 is a relation from A to C via B in a sense. If R1 is a parent-child relation and R2 is a sister relation, then R1R2 is an aunt-nephew/niece relation.

Example 1: Let A = {a1 , a2} , B = {b1 , b2 , b3} , and C = {c1 , c2} . Also let R1 = {<a1 , b1>,<a1 , b2>,<a2 , b3>} , and R2 = {<b1 , c1>,<b2 , c1>,<b2 , c2>,<b3 , c1>} . Then R1R2 = {<a1 , c1>,<a1 , c2>,<a2 , c1>} .

This is illustrated in Figure 6. The dashed lines in the figure of R1R2 indicate the ordered pairs in R1R2, and dotted lines show ordered pairs that produce the dashed lines. (The lines in the left figure are all supposed to be solid lines.)