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Then the converse of S is the statement Q implies P (Q → P). In general, the truth of S says nothing about the truth of its converse, [2] unless the antecedent P and the consequent Q are logically equivalent. For example, consider the true statement "If I am a human, then I am mortal."
Using composition of relations, the converse may be composed with the original relation. For example, the subset relation composed with its converse is always the universal relation: ∀A ∀B ∅ ⊂ A ∩B ⇔ A ⊃ ∅ ⊂ B ⇔ A ⊃ ⊂ B. Similarly, For U = universe, A ∪ B ⊂ U ⇔ A ⊂ U ⊃ B ⇔ A ⊂ ⊃ B.
For example, "is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x. Antisymmetric for all x, y ∈ X, if xRy and yRx then x = y. For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false). [11] Asymmetric
In logic and mathematics, statements and are said to be logically equivalent if they have the same truth value in every model. [1] The logical equivalence of p {\displaystyle p} and q {\displaystyle q} is sometimes expressed as p ≡ q {\displaystyle p\equiv q} , p :: q {\displaystyle p::q} , E p q {\displaystyle {\textsf {E}}pq} , or p q ...
Hypernymy and hyponymy are converse relations. If X is a kind of Y, then X is a hyponym of Y and Y is a hypernym of X. [7] Hyponymy is a transitive relation: if X is a hyponym of Y, and Y is a hyponym of Z, then X is a hyponym of Z. [8] For example, violet is a hyponym of purple and purple is a hyponym of color; therefore violet is a hyponym of ...
In mathematical logic, a theory can be extended with new constants or function names under certain conditions with assurance that the extension will introduce no contradiction. Extension by definitions is perhaps the best-known approach, but it requires unique existence of an object with the desired property. Addition of new names can also be ...
A model of a theory that satisfies the axioms of the theory but has properties not intended by the original formulation, often revealing the theory's consistency or independence results. normal form In logic, a standardized way of expressing logical formulas, such as conjunctive normal form (CNF) or disjunctive normal form (DNF), to facilitate ...
In logic, converse nonimplication [1] is a logical connective which is the negation of converse implication (equivalently, the negation of the converse of implication). Definition [ edit ]