In this introductory chapter some mathematical notions are presented rapidly
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implication, where is the 'hypothesis' or 'assumption', the 'consequence'
or 'conclusion'. By definition, the formula is false if p is true and false, otherwise it is always true. In other words the implication does not allow to deduce a false conclusion from a true assumption, yet does not exclude a true conclusion being implied by a false hypothesis. Thus the statement 'if it rains, I'll take the umbrella' prevents me from going out without umbrella when it rains, but will not interfere with my decision if the sky is clear. Using p and q it is easy to check that the formula has the same truth value of Therefore the connective can be expressed in terms of the basic connectives and Other frequent statements are structured as follows: 'the conclusion is true if and only if the assumption is true', or 'necessary and sufficient condition for a true is a true '. Statements of this kind correspond to the formula ( is 6 1 Basic notions (logically) equivalent to ), called logic equivalence. A logic equivalence is true if and are simultaneously true or simultaneously false, and false if the truth values of and differ. An example is the statement 'a natural number is odd if and only if its square is odd'. The formula is the conjuction of and in other words and ( ) ) have the same truth value. Thus the connective can be expressed by means of the basic connectives and . The formula p ⇒ q (SL statement like 'if , then ) can be expressed in various other forms, all logically equivalent. These represent rules of inference to attain the truth of the implication. For example, p ⇒ q is logically equivalent to the formula , called contrapositive formula; symbolically This is an easy check: is by definition false only when is true and false, i.e., when is true and false. But this corresponds precisely to the falsehood of ,. Therefore we have established the following inference rule: in order to prove that the truth of implies the truth of , one may assume that the conclusion is false and deduce from it the falsehood of the assumption . To prove for instance the implication 'if a natural number is odd, then 10 does not divide it', we may suppose that the given number is a multiple of 10 and (easily) deduce that the number must be even. A second inference rule is the so-called proof by contradiction, which we will sometimes use in the textbook. This is expressed by In order to prove the implication one can proceed as follows: suppose p is true and the conclusion is false, and try to prove the initial hypothesis false. Since is also true, we obtain a self-contradictory statement. A more general form of the proof by contradiction is given by the formula where is an additional formula: the implication is equivalent to assuming true and false, then deducing a simultaneously true and false statement (note that the formula is always false, whichever the truth value of r). Download 50.42 Kb. Do'stlaringiz bilan baham: |
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