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A proof by contrapositive is a direct proof of the contrapositive of a statement. [14] However, indirect methods such as proof by contradiction can also be used with contraposition, as, for example, in the proof of the irrationality of the square root of 2 .
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.
Since an inverse is the contrapositive of the converse, inverse and converse are logically equivalent to each other. [1] For example, substituting propositions in natural language for logical variables, the inverse of the following conditional proposition "If it's raining, then Sam will meet Jack at the movies." would be
It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument. The history of the inference rule modus tollens goes back to antiquity. [4]
Given a type A statement, "All S are P.", one can make the immediate inference that "All non-P are non-S" which is the contrapositive of the given statement. Given a type O statement, "Some S are not P.", one can make the immediate inference that "Some non-P are not non-S" which is the contrapositive of the given statement.
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.
For example, consider the true statement "If I am a human, then I am mortal." The converse of that statement is "If I am mortal, then I am a human," which is not necessarily true. However, the converse of a statement with mutually inclusive terms remains true, given the truth of the original proposition.
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 q / 2 is even smaller than q and still positive.