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In mathematics, proof by contrapositive, or proof by contraposition, is a rule of inference used in proofs, where one infers a conditional statement from its contrapositive. [15] In other words, the conclusion "if A, then B" is inferred by constructing a proof of the claim "if not B, then not A" instead. More often than not, this approach is ...
For example, going from "No S are P" to its converse "No P are S". In the words of Asa Mahan: "The original proposition is called the exposita; when converted, it is denominated the converse. Conversion is valid when, and only when, nothing is asserted in the converse which is not affirmed or implied in the exposita." [5]
The thing of importance is that the dog detects or does not detect an intruder, not whether there is one.) Example 1: If I am the burglar, then I can crack a safe. I cannot crack a safe. Therefore, I am not the burglar. Example 2: If Rex is a chicken, then he is a bird. Rex is not a bird. Therefore, Rex is not a chicken.
There are many places to live in California other than San Diego. On the other hand, one can affirm with certainty that "if someone does not live in California" (non-Q), then "this person does not live in San Diego" (non-P). This is the contrapositive of the first statement, and it must be true if and only if the original statement is true ...
A typical example is the proof of the proposition "there is no smallest positive rational number": assume there is a smallest positive rational number q and derive a contradiction by observing that ā q / 2 ā is even smaller than q and still positive.
The inverse and the converse of a conditional are logically equivalent to each other, just as the conditional and its contrapositive are logically equivalent to each other. [1] But the inverse of a conditional cannot be inferred from the conditional itself (e.g., the conditional might be true while its inverse might be false [2]). For example ...
A bijective (one-to-one and onto) correspondence between two structures that preserves the operations and relations of the structures, indicating they have the same form or structure. iteration The process of repeating a set of operations or a procedure multiple times, each time applying it to the result of the previous step. iteration theorem
Corner quotes, also called “Quine quotes”; for quasi-quotation, i.e. quoting specific context of unspecified (“variable”) expressions; [3] also used for denoting Gödel number; [4] for example “āGā” denotes the Gödel number of G. (Typographical note: although the quotes appears as a “pair” in unicode (231C and 231D), they ...