In propositional logic, transposition is a valid rule of replacement that permits one to switch the antecedent with the consequent of a conditional statement in a logical proof if they are also both negated. It is the inference from the truth of "A implies B" to the truth of "Not-B implies not-A", and conversely. It is very closely related to the rule of inference modus tollens. It is the rule that
where "" is a metalogical symbol representing "can be replaced in a proof with".
The transposition rule may be expressed as a sequent:
where is a metalogical symbol meaning that is a syntactic consequence of in some logical system;
or as a rule of inference:
where the rule is that wherever an instance of "" appears on a line of a proof, it can be replaced with "";
or as the statement of a truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as:
where and are propositions expressed in some formal system.
In the inferred proposition, the consequent is the contradictory of the antecedent in the original proposition, and the antecedent of the inferred proposition is the contradictory of the consequent of the original proposition. The symbol for material implication signifies the proposition as a hypothetical, or the "if-then" form, e.g. "if P then Q".
The biconditional statement of the rule of transposition (↔) refers to the relation between hypothetical (→) propositions, with each proposition including an antecent and consequential term. As a matter of logical inference, to transpose or convert the terms of one proposition requires the conversion of the terms of the propositions on both sides of the biconditional relationship. Meaning, to transpose or convert (P → Q) to (Q → P) requires that the other proposition, (~Q → ~P), be transposed or converted to (~P → ~Q).
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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. The contrapositive of a statement has its antecedent and consequent inverted and flipped. Conditional statement . In formulas: the contrapositive of is . If P, Then Q. — If not Q, Then not P. "If it is raining, then I wear my coat" — "If I don't wear my coat, then it isn't raining.
In propositional logic, double negation is the theorem that states that "If a statement is true, then it is not the case that the statement is not true." This is expressed by saying that a proposition A is logically equivalent to not (not-A), or by the formula A ≡ ~(~A) where the sign ≡ expresses logical equivalence and the sign ~ expresses negation. Like the law of the excluded middle, this principle is considered to be a law of thought in classical logic, but it is disallowed by intuitionistic logic.
In logic and mathematics, the converse of a categorical or implicational statement is the result of reversing its two constituent statements. For the implication P → Q, the converse is Q → P. For the categorical proposition All S are P, the converse is All P are S. Either way, the truth of the converse is generally independent from that of the original statement. Let S be a statement of the form P implies Q (P → Q). Then the converse of S is the statement Q implies P (Q → P).
Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics a