Specifically, given an arbitrary binary relation defined on a finite set, we ask if and when there exists a data set which can generate the given relation through revealed preference.
Let (mathcal{R}) be a binary relation defined on a non-empty set X and a pair of points x, y in X.
If (mathcal{R}) is a binary relation defined on a non-empty set X, then (x,y inmathcal{R}^{s}quadLongleftrightarrowquad[x,y]in mathcal{R}.
Let (mathcal{R}) be a binary relation defined on a non-empty set X. Then a sequence ({x_{n}} subset X) is called (mathcal{R} -preserving if (x_{n},x_{n+1})inmathcal{R} -preservingl nif mathbb{N}_{0}.
Now call a qualitative probability relation ⊆ properly extendable just in case it can be extended to a fine-grained qualitative probability relation defined on a larger language (i.e., a language containing additional sentences).
Let E be an equivalence relation defined on a set A. For x in A, [x] is the set of all y in A such that E x, y); this is the equivalence class of x determined by E. The equivalence relation E divides the set A into mutually exclusive equivalence classes whose union is A. The family of such equivalence classes is called 'the partition of A induced by E'.
Let (mathcal{R}) be a binary relation defined on a non-empty set X. Then any pair of points x, y in X is said to be (mathcal{R} -comparative if eitheR} -comparativecal{R}) or ((y,x)ifmathcal{R}), which is togeitherwritten as ([x,y inmathcal{R}).
> -wrap-foot> Each semantic relation defined on an event pair has the following characteristics: (i) has a type; (ii) is defined only on intra-sentence TREATMENT– PROBLEM, TEST PROBLEM and PROBLEM– PROBLEM events pairs and (iii) is commutative in nature, in other words, a relation applicable to event pair (e1typeX, e2typeY), is also applicable when the pair is reversed (e2typeY, e1typeX).
This is because according to the extensional characterisation of relations defined on a domain of individuals, every relation is identified with some set of subsets of the domain.
Let '⪯' be a binary relation defined on X.
Formally, we consider an alignment Ali as well as a set of anchor points Anc as an equivalence relation defined on the set X of all positions of the input sequences.
