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A relation is reflexive if, and only if, its complement is irreflexive. A relation is strongly connected if, and only if, it is connected and reflexive. A relation is equal to its converse if, and only if, it is symmetric. A relation is connected if, and only if, its complement is anti-symmetric.
Sometimes used for “relation”, also used for denoting various ad hoc relations (for example, for denoting “witnessing” in the context of Rosser's trick). The fish hook is also used as strict implication by C.I.Lewis p {\displaystyle p} ⥽ q ≡ ( p → q ) {\displaystyle q\equiv \Box (p\rightarrow q)} .
A relation algebra (L, ∧, ∨, −, 0, 1, •, I, ˘) is an algebraic structure equipped with the Boolean operations of conjunction x∧y, disjunction x∨y, and negation x −, the Boolean constants 0 and 1, the relational operations of composition x•y and converse x˘, and the relational constant I, such that these operations and constants satisfy certain equations constituting an ...
def – define or definition. deg – degree of a polynomial, or other recursively-defined objects such as well-formed formulas. (Also written as ∂.) del – del, a differential operator. (Also written as.) det – determinant of a matrix or linear transformation. DFT – discrete Fourier transform.
A relation R is called intransitive if it is not transitive, that is, if xRy and yRz, but not xRz, for some x, y, z. In contrast, a relation R is called antitransitive if xRy and yRz always implies that xRz does not hold. For example, the relation defined by xRy if xy is an even number is intransitive, [13] but not antitransitive. [14]
These imprecise uses of the word regular are not to be confused with the notion of a regular topological space, which is rigorously defined. resp. (Respectively) A convention to shorten parallel expositions. "A (resp. B) [has some relationship to] X (resp. Y)" means that A [has some relationship to] X and also that B [has (the same ...
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The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by x R the equivalence class of x, consisting of all elements z such that x R z. Then the relation x R y is equivalent with the equality x R = y R.