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Mathematics LibreTexts

17.2: Operations on Relations

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Viewing relations as subsets of Cartesian products suggests ways to build new relations from old.

Definition: Union (of relations R1,R2)

the relation where a(R1R2)b means that at least one of aR1b or aR2b is true

Definition: Intersection (of relations R1,R2)

the relation where a(R1R2)b means that both aR1b and aR2b are true

Definition: Complement (of relation R)

the relation where aRCb means that aRb is not true

Definition: a  b

alternative notation for aRCb

Note 17.2.1

Considering relations as subsets of Cartesian products, the above relation operations mean precisely the same thing as the corresponding set operations.

Example 17.2.1: Union of “less than” and “equal to” relations.

Consider the relations < and = on R, and let R be the union <=. Then xRy means that at least one of x<y or x=y is true. That is, R is the same as the relation .

Example 17.2.2: Sibling relations.

Let H represent the set of all living humans. Let relations RF,RMH×H be defined by

  • aRFb if a,b have the same father; and
  • aRMb if a,b have the same mother.

Set RP=RFRM. Then aRPb means that a,b have the same parents.

Example 17.2.3: Complement of the subset relation.

Let U be a universal set and consider the relation on P(U). Then ACB means that A is not a subset of B, which can only happen if some elements of A are not in B. In other words, ACB means that ABC.

Careful.

Relation ACB does not (necessarily) mean ABC. Draw a representative Venn diagram to see why.

Unlike functions, which can only be reversed if bijective, every relation can be reversed by simply stating the relationship in the reverse order.

Definition: Inverse (of a relation R)

the relation where bR1a means that aRb is true

Note 17.2.2

  • As subsets of Cartesian products, if RA×B, then R1B×A, and (a,b)R if and only if (b,a)R1.
  • A relation R and its inverse R1 express the same relationship between elements of two sets A and B, just phrased in the opposite order. In logical terms, bR1aaRb.

Example 17.2.4: Parent/child relations.

Let H represent the set of all living humans, and let R represent the relation on H where h1Rh2 means human h1 is the parent of human h2. Then h2R1h1 means human h2 is the child of human h1. Both relations express the same information, but in a different order.

Example 17.2.5: Inverse of Division Relation.

Recall that is a relation on N>0 where mn means that m divides n. Then for the inverse relation, n1m means n is a multiple of m. Both relations express the same information, but in a different order.

Example 17.2.6: Inverse of Logical Equivalence

Let L represent the set of all possible logical statements. We have a relation on L, where AB means that logical statement A involves the same statement variables and has the same truth table as logical statement B. Since AB if and only if BA, we conclude that the logical equivalence relation on L is its own inverse.

There are two more set-theoretic ideas we can reinterpret as relations.

Definition: Empty Relation

the relation between sets A and B corresponding to the empty subset A×B, so that ab is always false

Definition: Universal Relation

the relation between sets A and B corresponding to the full subset U=A×BA×B, so that aUb is always true.


This page titled 17.2: Operations on Relations is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.

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