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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 .
For example, there exist simple algebraic expressions for angle bisectors in terms of the sides of the triangle. Equating two of these expressions and algebraically manipulating the equation results in a product of two factors which equal 0, but only one of them ( a − b ) can equal 0 and the other must be positive.
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]
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 x {\displaystyle x} , if x 2 {\displaystyle x^{2}} is even, then x {\displaystyle x} is even:
This led to a natural curiosity with regards to geometry and trigonometry – particularly triangles and rectangles. These were the shapes which provided the most questions in terms of practical things, so early geometrical concepts were focused on these shapes, for example, the likes of buildings and pyramids used these shapes in abundance.
MEXICO CITY (Reuters) -Mexico is seeking an agreement with U.S. President-elect Donald Trump to ensure Mexico does not receive deportees from third countries in case of large-scale deportations ...
Such a proof is again a refutation by contradiction. 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.
As the holidays approach, many people get curious about reindeer. Have you ever wondered how fast they can run? Well, they may not be able to pull a sleigh around the world in a single night, but ...