Prologue to Proofs .


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2/9/2004. Discrete Mathematics for Teachers, UT Math 504, Lecture 05. 2. Foundation . The guidelines of chess characterize the diversion. You counsel them to figure out what moves are lawful and whether a position closes in a win for White, a win for Black, or a draw. In the meantime they let you know nothing about how to play chess. They figure out which moves are legitimate yet not which moves are great, which strategies indisp
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Slide 1

Prologue to Proofs

Slide 2

Background The guidelines of chess characterize the amusement. You counsel them to figure out what moves are legitimate and whether a position closes in a win for White, a win for Black, or a draw. In the meantime they reveal to you nothing about how to play chess. They figure out which moves are legitimate however not which moves are great, which strategies irreplaceable, which techniques successful, and which moves delightful. They distinguish winning positions yet offer no arrangements for contacting them. New players must take in the guidelines until they turn out to be second nature and tail them unfailingly, however the players\' judgment and innovativeness, sharpened by steady review, prompt to excellent triumphs. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

Slide 3

Background Propositional rationale gives the principles to demonstrating scientific truths. They figure out if derivation is honest to goodness yet not whether it is helpful, viable, or enhance. They distinguish substantial contentions however offer no arrangements for defining them. New mathematicians must take in the principles until they turn out to be second nature and tail them unfailingly, yet the mathematicians\' judgment and innovativeness, sharpened by constant review, prompt to wonderful confirmations. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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Background We are practically prepared to cutting our teeth on a few hypotheses and evidences. The staying preparatory is a concise talk of the predicate math . Most hypotheses are not explanations about individual cases (like ­"4­ 2 is even") yet rather asserts that specific recommendations hold all around (like "the square of each considerably number is even") or articulations about presence or non-presence of cases (like "each whole number more than 1 has a prime component" or "there is no biggest prime number"). The predicate analytics shows us how to deal with such general and existential cases. When we move beyond the predicate analytics, we will begin composing proofs. We will start with various standard ways to deal with verifications, trailed via cautious examination of the specific evidence system called scientific acceptance. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus A recipe is a number juggling expression with factors in it. We can allot qualities to ( instantiate ) the factors from some endless supply of passable numbers. When we do as such the equation turns into a number. For example the equation x+3 is not a number, but rather when we instantiate x by setting x=5, for example, the recipe turns into the number 8. A recipe is, truth be told, a capacity from some area of numbers into some codomain of numbers. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus A predicate is an English expression as a recommendation yet with factors in it. We can instantiate the factors from some endless supply of conceivable qualities (called the universe of talk or the area of talk ). When we do as such, the predicate turns into a suggestion. For example assume we consider the announcement, "x is a University of Tennessee science educator," where the universe of talk for x is the arrangement of individuals right now living on the planet. This announcement is a predicate. It has no truth esteem the length of x remains a variable. On the off chance that we instantiate x, in any case, by setting x=Reid Davis, then the predicate turns into a suggestion with a truth esteem. A predicate is, truth be told, a capacity from the universe of talk (space) into some codomain of recommendations. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus truth be told, the term propositional capacity is an equivalent word for predicate (it is a capacity that yields recommendations, much the same as a mind boggling capacity yields complex numbers, a levelheaded capacity yields sound numbers, and so forth.). We can indicate a predicate utilizing capacity documentation, for example characterizing P(x)= "x is a University of Tennessee science teacher." The P(Reid Davis) is valid, and P(Queen Elizabeth II) is false. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus Predicates can have more factors. For example we can frame the predicate P(x,y)="x is y years old" where the universe of talk for x is living individuals and for y is nonnegative whole numbers. At that point P(Reid Davis,29) is the false recommendation, "Reid Davis is 29 years of age." Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus Consider the predicate P(x)="the square of x is nonnegative" where the universe of talk for x is the arrangement of genuine numbers. we can create recommendations by instantiating x. For example, P(5) is the genuine recommendation, "the square of 5 is nonnegative." There are limitlessly numerous decisions for x and consequently interminably various suggestions P(x). They are, be that as it may, all genuine. In this manner the announcement "for each estimation of x, the suggestion P(x) is valid" is a genuine recommendation. This delineates another approach to transform a predicate into a recommendation: We affirm reality of the suggestion for all estimations of the variable. This is one method for evaluating a variable. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus There are just two quantifiers in like manner numerical use. The one we just examined is the all inclusive quantifier , which affirms reality of a predicate for each estimation of a variable. The documentation for general evaluation is ∀ , a topsy turvy and in reverse A. In this way one composes ∀ x P(x), which is perused, "for all x, P(x)" or "for each x, P(x)." For example one may compose, ∀ x the square of x is nonnegative, which is perused, "for all x, the square x is nonnegative" or "the square of each x is nonnegative" or even "the square of each genuine x is nonnegative" (determining the universe of talk). Discrete Mathematics for Teachers, UT Math 504, Lecture 05

Slide 11

The Predicate Calculus The other normal quantifier is the existential quantifier , which states the presence of an estimation of the variable that makes the predicate the predicate genuine. The documentation for all inclusive evaluation is ∃ , a topsy turvy and in reverse E. One composes ∃ x P(x), which is perused, "there exists a x to such an extent that P(x)" or "for some x, P(x)." For example one may compose, ∃ x the number x is even and prime, with the universe of talk for x being the positive whole numbers. One peruses this, "there exists a x with the end goal that the number x is even and prime" or "some positive whole number x is even and prime." This recommendation is valid since there is no less than one estimation of x (in particular x=2) that make the predicate genuine. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus A predicate turns into a suggestion when we instantiate or evaluate each of its factors. For example, let P(x,y,z) be the predicate x+y=z where the universe of talk for every one of the three factors is the whole numbers. At that point for example ∀ x  ∀ y P(x,y,3) is the false recommendation, "for each x and each y, x+y=3" We can negate a generally evaluated suggestion by finding a solitary instantiation that makes the predicate false. For this situation P(2,4,3) affirms 2+4=3, which is false. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus Similarly we can demonstrate an existentially evaluated suggestion by finding a solitary instantiation that makes the predicate genuine. With P(x,y,z)="x+y=z", the recommendation ∃ x  ∃ y P(x,y,3) states, "There are estimations of x and y for which x+y=3." This is valid since P(1,2,3) is valid. Request of evaluation does not make a difference the length of all quantifiers are of a similar sort (all general or all existential). For example ∀ x  ∀ y and ∀ y  ∀x create a similar suggestion, as do ∃x ∃y and ∃y ∃x. In the event that we blend all inclusive and existential quantifiers, be that as it may, then the request is significant. For example ∀x ∃y y>x says, "for each x there exists a y with the end goal that y>x." as such, regardless of what number we pick, there is a bigger number. This is valid since, for example, we can simply utilize y=x+1. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus On the other hand, ∃x ∀y y>x says, "there is a x with the end goal that for each y the case y>x." as it were there is a solitary x that is littler than each y." This is doubtlessly false since paying little mind to what x we pick, the esteem y=x−1 will be littler. It is some of the time hard to peruse suggestions accurately when they have blended quantifiers, so dependably approach such issues painstakingly. Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus Negating an all around evaluated predicate is legitimately equal to existentially measuring the nullified predicate. That is ~∀x P(x) ≡ ∃x ~P(x). Any illustration makes this self-evident. For example saying, "it is not that case that each prime is odd" is comparable to stating, "there is a prime that is not odd." Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus Negating an existentially evaluated predicate is legitimately proportionate to all around measuring the refuted predicate. That is ~∃x P(x) ≡ ∀x ~P(x). This is likewise clear from a case. For example, "there is no prime number that is the biggest prime," is unmistakably equal to, "each prime number is not the biggest prime." Discrete Mathematics for Teachers, UT Math 504, Lecture 05

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The Predicate Calculus This control likewise works for numerous quantifiers. The straightforward decide is that when a refutation "goes through" a quantifier, the quantifier changes to the next quantifier. For example (in a truncated nota

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