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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.
An indirectly self-referential sentence would replace the phrase "this sentence" with an expression that effectively still referred to the sentence, but did not use the pronoun "this." An example will help to explain this. Suppose we define the quine of a phrase to be the quotation of the phrase followed by the phrase itself. So, the quine of:
An indirect statement or question can replace the direct object of a verb that is related to thought or communication. An indirect statement is expressed by changing the case of the subject noun phrase from nominative to accusative and by replacing the main verb with an infinitive (as in the English phrase "You believe me to be a traitor" above).
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. By the definition of a rational number , the statement can be made that " If 2 {\displaystyle {\sqrt {2}}} is rational, then it can be expressed as an irreducible fraction ".
A formal expression is a kind of string of symbols, created by the same production rules as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two formal expressions are considered equal only if they are syntactically equal, that is, if they are the exact same expression.
Quintic function: Fifth degree polynomial. Rational functions: A ratio of two polynomials. nth root. Square root: Yields a number whose square is the given one. Cube root: Yields a number whose cube is the given one.
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