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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.
Proof by contraposition infers the statement "if p then q" by establishing the logically equivalent contrapositive statement: "if not q then not p". For example, contraposition can be used to establish that, given an integer , if is even, then is even: Suppose is not even.
A system will be said to be inconsistent if it yields the assertion of the unmodified variable p [S in the Newman and Nagel examples]. In other words, the notion of "contradiction" can be dispensed when constructing a proof of consistency; what replaces it is the notion of "mutually exclusive and exhaustive" classes.
[3] A mathematical proof employing proof by contradiction usually proceeds as follows: The proposition to be proved is P. We assume P to be false, i.e., we assume ¬P. It is then shown that ¬P implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, Q and ¬Q, and appealing to the law of ...
Each proposition has three essential parts: a subject, a predicate, and a copula connecting the subject to the predicate. [107] For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is". [108] The subject and the predicate are the terms of the proposition. Aristotelian logic ...
This is the contrapositive of the first statement, and it must be true if and only if the original statement is true. Example 2. If an animal is a dog, then it has four legs. My cat has four legs. Therefore, my cat is a dog.
A three-valued logic where the third truth value is the truth-value gap "neither true nor false" ("N"), and the designated values are "true" and "neither true nor false." [10] analysis 1. Analysis, the process of breaking a concept down into more simple parts, so that its logical structure is displayed. 2. Mathematical analysis analytic
The form of a modus tollens argument is a mixed hypothetical syllogism, with two premises and a conclusion: . If P, then Q. Not Q. Therefore, not P.. The first premise is a conditional ("if-then") claim, such as P implies Q.