H relation

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H relations are used in mathematically describing interaction among people.

Let A and B be sets and let A × B = {(a, b) | a belongs to A and b belongs to B} be their Cartesian product.

Definition: If R is a subset of A × B, then we call R a relation on A × B.

Definition: Let X be a set and a subset R of X × X a relation on X (actually on X × X), suppose that R strongly fails to be transitive, that is, for all a, b, c in X, if (a, b) is in R and (b, c) is in R, then (a, c) is not in R. In this case, we say that R is antitransitive on X and we say that R is an "H relation on X".

Notation: We will use the letter H to represent H relations and write aHb for (a, b) belongs to H.

A truth table with 8 possible combinations of truth values for aHb,bHc, and aHc shows that we may characterize antitransitivity by: For all a,b,c in X, not[(aHb)and(bHc)and(aHc)].

Definition: In a similar manner, a relation R is said to be antireflexive if it is "reflexive nowhere": For all a in X, (a,a) is not in R.

Note: No condition concerning symmetric relations is considered here. H relations reflect the rule, "The enemy of your enemy may be your ally." In fact, one is not his own enemy:

Proposition: If H is anititransitive, then H is antireflexive.

Proof (by contrapositive): Suppose that H is not antireflexive, that is, there exists some a in X with aHa. Then (aHa)and(aHa)and(aHa), contradicting the condition for H being antitransitive when a = b = c.

Examples: Let X be a group of 12 people seated around a circle.

Let R = {(a,b)|b is older than a}, then R is transitive since b is older than a and c is older than b implies that c is older than a.

Let H₁ = {(a, b)| b sits immediately to the left of a}. Then H₁ is antitransitive (and antireflexive) on X because aHb and bHc implies that b occupies the seat between a and c, so c is not seated immediately to the left of a.