## Operations on Relations

### Subjects to be Learned

• set operations on relation
• composite relation
• properties of composite relation
• powers of relation

### Contents

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 the following figure. 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.)

Example 2: If R is the parent-child relation on a set of people A, then RR, also denoted by R2, is the grandparent-grandchild relation on A.

More examples:
The digraphs of R2 for several simple relations R are shown below:

Properties of Composite Relations

Composite relations defined above have the following properties. Let R1 be a relation from A to B, and R2 and R3 be relations from B to C. Then

1. R1(R2R3) = (R1R2)R3
2. R1(R2 R3) = R1R2 R1R3
3. R1(R2 R3) R1R2 R1R3

Proofs for these properties are omitted.

Powers of Relation

Let R be a binary relation on A. Then Rn for all positive integers n is defined recursively as follows:

Definition(power of relation):
Basis Clause: R0 = E, where E is the equality relation on A.
Inductive Clause: For an arbitrary natural number n , Rn+1 = RnR.
Note that there is no need for extremal clause here.

Thus for example R1 = R, R2 = RR, and R3 = R2R = (RR)R = R(RR) = RRR.

The powers of binary relation R on a set A defined above have the following properties.

1. Rm+n = RmRn,
2. (Rm)n = Rmn.

### Test Your Understanding of Properties of Operations on Relations

Indicate which of the following statements are correct and which are not.
Click True or False , then Submit. There are two sets of questions.

Next -- Closures of Binary Relation

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