Thomas A. Alspaugh

Relations

A (binary) relation R between sets X and Y is a subset of X×Y. (X×Y is a Cartesian product.)

Thus, a relation is a set of pairs.

The interpretation of this subset is that it contains all the pairs for which the relation is true. We write xRy if the relation is true for x and y (equivalently, if (x,y)∈R).

X and Y can be the same set, in which case the relation is said to be "on" rather than "between":

A (binary) relation R on set E is a subset of E×E. (E×E is a Cartesian product.)

Examples using E={0,1,2,3}:

- {(0,0), (1,1), (2,2), (3,3)}. This relation is =.
- {(0,1), (0,2), (0,3), (1,2), (1,3), (2,3)}. This relation is <.
- {(0,0), (1,1), (1,0), (2,2), (2,1), (2,0), (3,3), (3,2), (3,1), (3,0)}. This relation is ≥.

Relations may also be of other arities.
An n-ary relation
R between sets
X_{1}, … ,
and
X_{n}
is a subset of the
n-ary product
X_{1}×…×X_{n},
in which case R is a set of
n-tuples.

The empty relation between sets X and Y, or on E, is the empty set ∅.

The empty relation is false for all pairs.

The full relation (or universal relation) between sets X and Y is the set X×Y.

The full relation on set E is the set E×E.

The full relation is true for all pairs.

The identity relation on set E is the set {(x,x) | x∈E}.

The identity relation is true for all pairs whose first and second element are identical.

The three relations below are possible definitions of the relation "likes" on the set {Ann, Bob, Chip}.

- Happy world
- In this world, "likes" is the full relation on the universe. Because it is full, every person likes every other person, including him- or herself. The extension of this "likes" relation is {(Ann,Ann),(Ann,Bob),(Ann,Chip),(Bob,Ann),(Bob,Bob),(Bob,Chip), (Chip,Ann),(Chip,Bob),(Chip,Chip)}.
- Narcissistic world
- In this world, "likes" is the identity relation on the universe. Because it is the identity relation, every person likes him- or herself, but no one else. The extension of this "likes" relation is {(Ann,Ann),(Bob,Bob),(Chip,Chip)}.
- Emptily unhappy world
- In this world, "likes" is the empty relation. Because it is empty, no person likes any other person, including him- or herself. The extension of this "likes" relation is {}.

Let R be a relation on E, and let x,y,z∈E.

A relation R is … | if … | A relation R is … | if … | |
---|---|---|---|---|

reflexive | xRx | irreflexive | xRy implies x≠y | |

symmetric | xRy implies yRx | antisymmetric | xRy and yRx implies x=y | |

transitive | xRy and yRz implies xRz |

Examples using =, <, and ≤ on integers:

- = is reflexive (2=2)
- = is symmetric (x=2 implies 2=x)
- < is transitive (2<3 and 3<5 implies 2<5)
- < is irreflexive (2<3 implies 2≠3)
- ≤ is antisymmetric (x≤y and y≤x implies x=y)

Note that while a relationship cannot be both
reflexive and irreflexive,
a relationship *can* be both symmetric and antisymmetric.
The = relationship is an example
(x=2 implies 2=x, and x=2 and 2=x implies x=2).

Examples using Ann, Bob, and Chip:

- Happy world
- "likes" is reflexive, symmetric, and transitive. (It is an equivalence relation.)
- Narcissistic world
- "likes" is reflexive, symmetric, antisymmetric, and transitive. (It is both an equivalence relation and a non-strict order relation, and on this world produces an antichain.)
- Emptily unhappy world
- "likes" is not reflexive, and is trivially irreflexive, symmetric, antisymmetric, and transitive.
- Circularly unhappy world
- This world's extension of "likes" is {(Ann,Bob),(Bob,Chip),(Chip,Ann)}. "likes" is irreflexive, antisymmetric, and not transitive.
- Somewhat-happy world with 2-circuit
- This world's extension of "likes" is {(Ann,Bob),(Bob,Ann),(Chip,Chip)}. "likes" is neither reflexive nor irreflexive, but is symmetric.
- Happy world with 2-circuit
- This world's extension of "likes" is {(Ann,Ann),(Ann,Bob),(Bob,Ann),(Bob,Bob),(Chip,Chip)}. "likes" is reflexive, symmetric, and transitive. (It is an equivalence relation.)

An equivalence relation is a relation that is reflexive, symmetric, and transitive.

Example: = is an equivalence relation, because = is reflexive, symmetric, and transitive.

An equivalence relation partitions its domain E into disjoint equivalence classes.

- Each equivalence class contains a set of elements of E that are equivalent to each other, and all elements of E equivalent to any element of the equivalence class are members of the equivalence class.
- The equivalence classes are disjoint: there is no x∈E such that x is in more than one equivalence class.
- The equivalence classes exhaust E: there is no x∈E such that x is in no equivalence class.
- Any element of an equivalence class may be its representative; the representative stands for all the members of its equivalence class.

Examples:

- Smaller circle plus dot happy world
- This world's "likes" is an equivalence relation, and so it induces a partition of {Ann,Bob,Chip}. The partition consists of two equivalence classes, {Ann,Bob} and {Chip}. These equivalence classes are disjoint (their intersection is empty) and exhaustive (every element is in an equivalence class).
- Narcissistic world
- This world's "likes" is also an equivalence relation, inducing a partition consisting of three singleton equivalence classes {Ann}, {Bob}, and {Chip} (clearly disjoint and exhaustive).
- Happy world
- This world's "likes" is also an equivalence relation. The partition it induces consists of one equivalence class, {Ann,Bob,Chip} (exhaustive and trivially disjoint).

An order (or partial order) is a relation that is antisymmetric and transitive.

Examples:

- ≤ is an order relation on numbers.
- ⊆ is an order relation on sets.
- The prefix relation on binary strings is an order relation.
- The symbol ⊑ is often used to represent an arbitrary partial order.

In mathematics and formal reasoning, order relations are commonly allowed to include equal elements as well. However, for some authors and in everyday usage, orders are more commonly irreflexive, so that "John is taller than Thomas" does not include the possibility that John and Thomas are the same height.

A strict order is one that is irreflexive and transitive; such an order is also trivially antisymmetric because there is no x and y such that xRy and yRx.

A non-strict order is one that is reflexive, antisymmetric, and transitive. Any order we discuss will be considered non-strict unless specifically stated otherwise.

Examples:

- < is strict, ≤ is non-strict.
- ⊂ is strict, ⊆ is non-strict.
- "taller than" is strict (no one is taller than him- or herself).

An order relation R on E is a total order if either xRy or yRx for every pair of elements x,y∈E.

An order relation R on E is a partial order if there is a pair of elements x,y∈E for which neither xRy nor yRx.

Let R be an order relation on E and let x,y∈E. x and y are incomparable under R if neither xRy nor yRx. We write this as x||y (or x#y). From the definitions, we can see that a total order is one for which no two elements are incomparable, and a partial order is one for which at least two elements are incomparable.

Examples:

- < on the integers is a total order. For any two integers x and y, either x<y or y<x.
- ⊂ on the powerset of the integers is a partial order; it is not total. There are many incomparable sets in this (and any other) powerset. For example, {1,4,9}⊄{1,3,5} and {1,3,5}⊄{1,4,9}, so these two sets are incomparable and we write {1,4,9}||{1,3,5} or {1,4,9}#{1,3,5}.

Let R be a relation from X to Y and S be a relation from Y to Z.

The composition (or join) of R and S, written R.S, is the relation {(x,z)∈X×Z | xRy and ySz for some y∈Y}.

A function-style notation S○R is also sometimes seen, but is quite inconvenient for relations. The notation R.S is easier to deal with as the relations are named in the order that leaves them adjacent to the elements that they apply to (thus x(R.S)z because xRy and ySz for some y).

The product of two relations R and S is the relation {(w,x,y,z) | wRx∧yRz} }

The converse
(or transpose)
of R,
written R^{−1},
is the relation
{(y,x) | xRy}.

The closure of a relation R is the relation {(x,z) | (x,y)∈R∧(y,z)∈R}.

The (reflexive) transitive closure of R is the smallest transitive relation S such that R⊆S. You can obtain the transitive closure of R by closing it, closing the result, and continuing to close the result of the previous closure until no further tuples are added.

Because relations are sets (of pairs), all the operations on sets also apply to relations.

The intersection of R and S, written RS, is the relation {x(RS)y | xRy and xSy}.

The union of R and S, written R∪S, is the relation {x(R∪S)y | xRy or xSy}.

The difference of R and S, written R−S or R \ S, is the relation {x(R−S)y | xRy but not xSy}.

Let

- X={Airplane,Pool,Restaurant}
- Y={Goggles,Heels,Seatbelt,Tuxedo}
- Z={Buckle,Pocket,Strap}
- R be the relation "is where one wears", with extension {(Airplane,Seatbelt),(Airplane,Tuxedo),(Pool,Goggles),(Restaurant,Heels),(Restaurant,Tuxedo)}
- S the relation "can have a", with extension {(Goggles,Strap),(Heels,Buckle),(Heels,Strap),(Seatbelt,Buckle),(Seatbelt,Strap),(Tuxedo,Pocket)}

- Composition
- R.S is {(Airplane,Buckle), (Airplane,Pocket), (Airplane,Strap), (Pool,Buckle), (Pool,Strap), (Restaurant,Buckle), (Restaurant,Pocket), (Restaurant,Strap)}.
- Converse
- R
^{-1}is {(Goggles,Pool),(Heels,Restaurant),(Seatbelt,Airplane),(Tuxedo,Airplane),(Tuxedo,Restaurant)}.

Transitivity and composition may seem similar: both are defined using x, y, and z, for one thing. But they are unrelated: transitivity is a property of a single relation, while composition is an operator on two relations that produces a third relation (which may or may not be transitive).

Symmetric and converse may also seem similar; both are described by swapping the order of pairs. But they are also unrelated: symmetry is a property of a single relation, while converse is an operator that takes a relation and produces another relation (which may or may not be symmetric). It is true, however, that the union of a relation with its converse is a symmetric relation.

Because relations are sets (of pairs), the relations on sets also apply to relations.

Let E be a set and R and S be relations on E.

R and S are equal if for every x,y∈E, xRy iff xSy.

R is a subset of S if for every x,y∈E, xRy implies xSy.

Examples:

- "Happy world" is equal to the full relation.
- "Smaller circle plus dot somewhat-happy world likes" is a subset of "Smaller circle plus dot happy world likes".

I thank Alex Fink and his unnamed student for pointing out an error in an earlier version of this page.