![]() ![]() " If the negation is true, then the original proposition (and by extension the contrapositive) is false. Negation (the logical complement), ¬ ( P → Q ) \neg (P\rightarrow Q) " It is not the case that if it is raining then I wear my coat.", or equivalently, " Sometimes, when it is raining, I don't wear my coat. Conversion (the converse), Q → P Q\rightarrow P "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). Inversion (the inverse), ¬ P → ¬ Q \neg P\rightarrow \neg Q "If it is not raining, then I don't wear my coat." Unlike the contrapositive, the inverse's truth value is not at all dependent on whether or not the original proposition was true, as evidenced here. The contrapositive ( ¬ Q → ¬ P \neg Q\rightarrow \neg P ) can be compared with three other statements: The law of contraposition says that a conditional statement is true if, and only if, its contrapositive is true. ![]() "If it is raining, then I wear my coat" - "If I don't wear my coat, then it isn't raining." In formulas: the contrapositive of P → Q P\rightarrow Q is ¬ Q → ¬ P \neg Q\rightarrow \neg P. The contrapositive of a statement has its antecedent and consequent inverted and flipped.Ĭonditional statement P → Q P\rightarrow Q. In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. For contraposition in the field of symbolic logic, see Transposition (logic). For contraposition in the field of traditional logic, see Contraposition (traditional logic). ![]()
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