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The converse may or may not be true, and even if true, the proof may be difficult. For example, the four-vertex theorem was proved in 1912, but its converse was proved only in 1997. [3] In practice, when determining the converse of a mathematical theorem, aspects of the antecedent may be taken as establishing context.
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.
Conversion (the converse), "If I wear my coat, then it is raining ." The converse is actually the contrapositive of the inverse, and so always has the same truth value as the inverse (which as stated earlier does not always share the same truth value as that of the original proposition).
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 ...
In logic, a set of symbols is commonly used to express logical representation. The following table lists many common symbols, together with their name, how they should be read out loud, and the related field of mathematics.
In mathematics and abstract algebra, a relation algebra is a residuated Boolean algebra expanded with an involution called converse, a unary operation.The motivating example of a relation algebra is the algebra 2 X 2 of all binary relations on a set X, that is, subsets of the cartesian square X 2, with R•S interpreted as the usual composition of binary relations R and S, and with the ...
In mathematical logic, geometric logic is an infinitary generalisation of coherent logic, a restriction of first-order logic due to Skolem that is proof-theoretically tractable. Geometric logic is capable of expressing many mathematical theories and has close connections to topos theory .
Mathematical logic, also called 'logistic', 'symbolic logic', the 'algebra of logic', and, more recently, simply 'formal logic', is the set of logical theories elaborated in the course of the nineteenth century with the aid of an artificial notation and a rigorously deductive method. [5]