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Stanford UniversityCourse
Introduction to Mathematical ThinkingPages
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2022
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Introduction to Mathematical Thinking. Tutorial for Assignment A: There is a natural number solution to the equation x cubed equals 27. (βπ₯βπ) π₯ 3 = 27 [ ] B: A million is not the largest natural number. There is an x in n such that x is greater than a million. (βπ₯βπ) π₯ > 1 000 000 [ ] C: The natural numbers are not all prime; there is a natural number p such that p > 1, and another natural number q such that q > 1, and n = pq. So there are naturalnumbers pq both of them greater than 1 such that n is their product. (βπβπ)(βπβπ) π > 1β§π > 1β§π = ππ [ ] Part A: In mathematics, relationships are expressed in a certain order of precedence, whichis why it's important to read formulas left to right. See how the formula " p in q "(where p is greater than one and q is greater than one) became the variable n ?That's because we must read things left to right, so we start with "there's a p in n"and finish with "there's a q in nβ. The equation x q = 28 cannot be solved using natural numbers. Β¬(βπ₯βπ) π₯ 3 = 28 [ ] The most obvious way of expressing this is to say that there is no natural number xthat satisfies the equation. Whereas to give an assertion in its canonical form is toexpress it in the form of a universal statement with an "all" quantifier at thebeginning. To say that for all x in N, it's not the case that x cubed equals 28, write inthis familiar fashion x cubed is not equal to 28. (βπ₯βπ) π₯ 3 β 28 [ ] The last part does not hold for x = 28. (βπ₯βπ) > π₯ 3 = 28 [ ] According to the statement, every natural number fails to be a solution. We can inferthat zero is less than every natural number because zero is a member of thesequence of integers, which begins with one and ends with infinity. π = 1, 2, 3β¦. { } Historically, the number zero was an unnatural number. In fact, originally, a zerosymbol was merely a circle that signified that there was nothing there, and aplaceholder was needed to denote that nothing existed in certain places when
performing calculations. This practice became common in India around 600 B.C.Today it is known that the number zero is less than every natural number. (βπ₯βπ) 0 < π₯ [ ] The statement "zero is less than every natural number" may be captured conciselyas "for all natural numbers x, x is greater than zero. π₯ > 0 [ ] There are several ways to express the statement that the natural number n is prime.The most important formula, however, is the one that captures the idea that it is notdivisible by any other natural number besides one and itself. (βπβπ)(βπβπ) (π = ππ)β(π = 1β¨π = 1) [ ] To say that a natural number, n, is prime requires one to first negate the universalquantifier statement. That is, if all possible factors were not equal to one of thenatural numbers itself, then it would be prime. The only way that two numbers can beequal to each other is if they are both 1. Therefore, because there are no numbersthat are multiplied by themselves (that is, the number 1), we can see that all possiblefactors of a number must be equal to 1. Therefore, if one of these factors is not equalto one, there is an error in the factorization somewhere. For every person x, there is a corresponding person y. So it's just that x loves y. βπ₯ ( ) βπ¦ ( )πΏ(π₯. π¦) For every pair of individuals, there is a pair they love. The individuals they love willdepend on the pair you start with. These individuals need not be the same for anytwo different starting pairs. Thus x loves y only if x and y are different people. Someforum participants misunderstood this point and were able to see the problem withthis statement only after it was explained; this shows that students need to learn howquantifiers work before they can understand why ordering is important. Part B: (βπ₯) ππππ(π₯)β¨πβπππ‘(π₯) [ ] For every person X, either X is tall or X is short. Everyone is one or the other. This isan example of an inclusive or, but of course, the properties themselves will make thisa disjoint. You want to have someone that is both tall and short although in real lifetall and short will have an overlap you know there comes a point where is someonetaller? Is someone short? So interpreting these is a matter of interpretation. Theheightβthis is a clear answer. The point is, things like this become ambiguous whenone considers that "tall" can mean different things. By writing something like this, wehave forced it to be precise. That is the whole point of what we are trying to do. Thisprecision is not necessarily the same precision that exists in the real world, but it isan interpretation within mathematics of something in the real world.
Part C: (βπ₯)ππππ(π₯)β¨(βπ₯)πβπππ‘(π₯) The statement "every person is tall" is equivalent to the statement "there exists atleast one person who is tall". The statement "everyone is short" is equivalent to thestatement "all people are short", which can be rewritten as "for every x, if x is aperson then x is short". These brackets, which bind everything inside them, are verytight. This means that you have to use parentheses if you want to use a quantifierthat binds everything like we did here. So in order to make this piece of text bindthese two words you would need to include a set of parentheses after "these". Weuse the word βbindsβ to describe quantifiers. That means it governs the xs in there.And the, this, the, the formal terminology of that is βbinds.β And so, in the bindingrules quantifiers bind everything that comes next to them, and if you want them tobind a disjunction, you have to put them in parentheses. So that has to become aunit to be bound through all, same would be true for exists. In this case, you don't need parentheses because all you've got is one predicate (tallx) , so you've got for all x tall x or for all x short x. You might ask yourself if you needparentheses here, and the answer to you in no. Generally, the rule for parentheses isyou put them in when you need them to disambiguate, but because this is the firsttime we're running through these. Though these statements seem different, they are both true. This statement is talkingabout something that is highly unlikely to be true in most societies. This statement iscertainly true if we're prepared to say where the tall and short people change atsome point. Part D: βπ₯ ( )Β¬π΄π‘ β βπππ(π₯) Some people might have read the sentence "For every person x, x is not at home"as a universal quantification, that is, as a statement that applies to all cases. Β¬ βπ₯ ( )π΄π‘ β βπππ (π₯) It depends on how you interpret these sentences. You could say that the mostnatural way of translating this sentence is as a universal quantification: "There is noone at home." Or, you could say that it is better rendered as an existentialquantification: "There is no person who's home." Both renderings are correct. βπ₯ ( )Β¬π΄π‘ β βπππ(π₯) β Β¬ βπ₯ ( )π΄π‘ β βπππ (π₯) As we've seen, these two assertions are equivalent. It's just a matter of choice as towhich you think more accurately reflects the nuances of the English language. Part E: πΆππππ (π½πβπ)β(βπ₯)[πππππ π₯ ( )βπΏπππ£ππ π₯ ( )]
If John comes, then all the women will leave. This is an if statement, and then there'sa conclusion. And the antecedent of that if statement is John coming. So we've gotJohn coming; there's a consequence here: all the women leave. Our variable xranges over people so we have to say for every x if x is a woman then x leaves. Sothat's our way of saying all women leave. For all x if x is a woman then x leaves. Part F: βπ₯ ( )[πππ π₯ ( )β§πΆππππ π₯ ( )]β(βπ₯)[πππππ π₯ ( )βπΏπππ£ππ π₯ ( )] In this case, it's not a single person John, it's any old man, so we'll have to say ifthere is an x who is a man and who comes, then every woman leaves. This is thesame as the previous thing but instead of saying John comes, Iβm saying there is anx who is a man and who comes. vNotice that number 4 is merely about expressingthe statement formally, in this case using quantifiers that range over the set of reelsand the natural numbers. The equation x2 + a = 0 has at least one real root for anynumber a, for any real number a. βπβπ ( ) βπ₯βπ ( )[π₯ 2 + π = 0] For all real numbers a, there is a real number x that satisfies the equation. For anyreal number a, this is a universal quantifier; in other words, it's a quantifier thatapplies to all real numbers. Even though that comes at the end of the sentence,which is fine in English, mathematicians find it amusing. However, its placementcauses confusion for mathematicians when they read it because it causes them towonder why we use a quantifier that applies to all real numbers. For some values ofa, you will not get an x as we know it. For instance, if a is negative, then thisstatement is true. However, if you take the negative a's, the x that solves it dependson the value of a. So you have to have an a before you can find the x. One way ofreading this is to say that if you give any value for a and we will find an x dependingon the value of a that solves this equation. Quantifying the order of the expression is crucial to solving this problem. Part bactually brings it to the previous one, but except we're really talking about anynegative real number and this is actually going to be true. So there's a quantifier forany real number and we're going to capture that as follows, we've got the set of realnumbers so we have to say for all real numbers a, if it's the case that a is negativethen there is an x that satisfies the equation. For any real number a, if it happens thata is negative, then there is an x which solves the equation. In mathematics, it is critical to be precise. The order of mathematical symbols is usedto convey meaning: a comes before x because the a determines (or is necessary for)the value of x. Okay, part c, every real number is rational, so, every real number x. βπ₯βπ ( ) βπβπ ( ) βπβπ ( )[π = ππ₯β¨π =β ππ₯β¨π₯ = 0
What we're trying to say is that, for every statement, there is an equivalent statementthat is true or false. For example, this one is false and this one is true. The point isthat you can express things in mathematics precisely whether they're true or false;and sometimes you have to express them formally in order to determine whetherthey're true or false. However, it's a separate issue from whether they're true or falsein itself; truthfulness or falsity is a separate issue from whether you can express itformally. Every real number can be expressed as the quotient of two integers. This istrue because every real number can be written as the ratio of two integers, or it canbe written as the difference between two integers, or it can be 0. In the natural numbers, there are two operationsβaddition and multiplication. Theproperty of divisibility also exists in the natural numbers, but division is not alwayspossible. Since m and n can be written by themselves without involving division, it isan elegant way to avoid writing m/n. There is a formula that expresses therelationship between x and m and n in terms of the order of the quantifiers. If youknow the values of those three variables, you can solve for the remaining ones.However, if a negation sign is placed before part c, then this formula does not work. βπ₯βπ ( ) βπβπ ( ) βπβπ ( )[πβ ππ₯β§πβ β ππ₯] There is a rational number (a number that can be expressed as the ratio of twointegers) which satisfies the following property: for all pairs of natural numbers m andn, m divided by n is not equal to x and negative m divides n is not equal tox.Previously, we had this junction because we were talking about a positivething.There is an x in R. This will be the irrational number we are assumingexists.With the property that for all m in N, m does not equal x nor does m equalnegative x. M divided by n is not equal to x andβm divided by n is not equal to x. βπ¦βπ ( ) βπ₯βπ ( )[ π₯ > π¦ ( ) β§ βπβπ ( ) βπβπ ( ) πβ ππ₯ ( )] There are two ways to define irrational numbers. One way is to say that for all realnumbers R, for all real numbers y, there exists a real number z which is not equal tothe quotient m/n. And if you're trying to say that there's no allowed irrational number,you're really going up into the positive range. So what happens on the left of zero onthe real line is irrelevant. Thus, we do not need to consider negative numbersbecause we are considering all real numbers. However, we should note that there isno largest real number; every real number can be divided into a positive part and anegative part. Real numbers are what mathematicians call "algebraic", which meansthey can be manipulated with addition and multiplication but not subtraction ordivision. However, it's fair to say that this is really a statement about just the irrationalnumbers. Given any irrational number, there is always an even bigger irrationalnumber. Once we get into the realm of real numbers, it gets extremely complicatedbecause you have to say, given any real number r , if that number is irrational thenthere is an even bigger irrational number. βπ¦βπ ( )[ βπβπ ( ) βπβπ ( ) πβ ππ¦ ( ) β βπ₯βπ ( ) π₯ > π¦ ( ) β§ βπβπ ( ) βπβπ ( ) πβ ππ₯ ( ) [ ]]
There are two ways to interpret the sentence, βThere is an irrational number biggerthan every real number.β One way is as a general statement about irrationalnumbers. The other way is as a specific statement about all the rational numbersand all the irrational numbers combined. If we interpret it as the latter, then it isequivalent to saying that every real number has an irrational number bigger thanitβand this statement is true. Once we have a formal description, it is unambiguous.As long as we express it correctly, it is not ambiguous. It has exactly oneinterpretation. This has several interpretations. I mean, here are two different ones.They're equivalent. They're obviously equivalent, for very obvious reasons. But we'vecashed them out in different ways. Well question five is our old friend about domesticcars and fallen cars so C is a set of all cars, Dx means x is domestic, Mx means x isbadly made. βπ₯βπΆ ( )[π· π₯ ( )βπ π₯ ( )] If a car is domestic, then it is badly made. All foreign cars are badly made, but we donot have a predicate for foreign so we must take it to mean "not domestic". In thiscase, we replace C by negative C and then we get: For all x and if x is not domesticthen it is badly made. βπ₯βπΆ ( )[Β¬π· π₯ ( )βπ π₯ ( )] And by negating the quantifier in this way, we are able to use the fact that negationbinds very tightly. This enables us to apply the quantified statement to whateverfollows it. Once you understand how this works, you should be able to follow thesekinds of statements more easily. Part C: All badly made cars are domestic. βπ₯βπΆ ( )[π π₯ ( )βπ· π₯ ( )] For all cars, if the car is badly made, then it's domestic. This statement is ambiguousand can be understood in several ways. However, there is at least one case in whichit is not the case that a car is domestic and badly made. βπ₯βπΆ ( )[π· π₯ ( )β§Β¬π π₯ ( )] There is a car called the x, which is domestic and not badly made. When makingcomparisons between universal and existential quantifiers, we typically compare animplication or a conjunction with an existential quantifier. When making comparisonsbetween a universal quantifier and a conditional, we typically compare a conjunctionwith a universal quantifier. In order to do this successfully, it's important tounderstand what these phrases mean rather than relying on symbolic rules. βπ₯βπΆ ( )[Β¬π· π₯ ( )β§π π₯ ( )] Question six concerns the same kind of concepts we've already looked at in earlierquestions, including whether certain things are rational or ordered in a set and wherethey fall on the list of largest rational numbers. However, we're dealing with differentrestrictions this time. We are asked to use quantifiers for real numbers. We do havea symbol q of x, meaning x is rational. So we're going to get a different expression.
The focus on what we are trying to make precise has been put elsewhere. We don'thave to talk about things being in reality or whatever because that's all we've got. βπ₯βπ¦[ π₯ < π¦ ( )ββπ§ π π§ ( ) β§ π₯ < π§ < π¦ ( ) ( )] Now we do have a bracket here because there are many things that follow thatdepend on x and y. So for all pairs of real numbers x and y with x less than y, there isa z which is rational and lies between them. Reading left to right, for all pairs of realnumbers x and y with x less than y, there is a z which is rational and lies betweenthem. Now we didn't write "exists" or "there exists," because Q isn't a set. In thiscase we have an implication (which means x is rational). Sometimes we have sets,sometimes we have predicates. Sometimes we use quantifiers. Sometimes we usepredicate quantifiers. They are all different ways of getting formality and precisioninto statements. βπ‘βππΉ(π, π‘) β§ βπβπ‘πΉ(π, π‘) β§ Β¬βπβπ‘πΉ(π, π‘) Number seven, which is attributed to Abraham Lincoln, is this famous quotation:"You can fool all of the people some of the time; you can even fool some of thepeople all of the time. But you cannot fool all of the people all of the time. But in mathematics we read strictly left to right, and so the quantifiers must comefirst. So let's say f, x, t means you can fool a person Woops. P that should have beena P shouldn't it. Then let me just change that now. Okay. Alright Fpt meaning you canfool person p at time t. Then let's do these one by one. You may fool all of the peoplesome of the time. That means there are some times when you can fool all of thepeople, there are some times when you can't fool any of them. There are some timeswhen you can fool all of the peopleβthe point is taking this clause, "you may fool allof the people some of the time," and then putting a comma before "there are times."The point is that we're making an argument about this clause and how it relates toother clausesβthere's a comma here because this is not part of our main clause;this is part of a sub-clause or an explanation for our main clauseβthere are timeswhen you can't fool any of them: there were sometimes when you could fool all ofthem. The quantifiers some, every and some are used to express different situations.Some are captured with the phrase there exists at least one. Every can beexpressed as there exists exactly one or there exists no more than one. Some isusually used to mean at least two limited cases, whereas every is used to mean alllimited cases. Nowadays, it is generally accepted that the word "some" can mean "at least one" inEnglish. However, in mathematics, it is more efficient to focus on a single objectrather than multiple objects. The quantifier captures this idea. You can fool all of thepeople some of the time and some of the people all of the time means there aresome people who can be fooled all of the time, at least 1 person can be fooled all ofthe time. In this clause, see what's been captured that's important. In other words,you can argue about whether or not some of the people have been well captured byexistence. However, what's really going on here is that all of the time you'll be able tofind some people who got fooled. You can fool some people all of the time. You can
find some people who will believe anything; you can find others who will believeanything. Let's examine the last example, but it is important to note that the conjunction βbutβ ismerely another form of conjunctions. You cannot fool all the people all the time, andthis one is the easiest because you only need two words for this sentence: alls andtime. There are times when it is possible to fool all of the people, but there are othertimes when it is impossible to fool all of the people. This first clause suggests thatthere are some people who can always be fooled. There are some people who canbe fooled some of the time, but not all of the time; we sometimes succeed in foolingthese individuals. However, there are also some people who cannot be fooled atallβthey will never believe what we say unless they have proof. That's why it is agood exercise, because it will help you understand how mathematical formulae cancapture the kinds of things we say in the real world. And this statement is areal-world statement with some cultural significance. Let's move to number 8: According to statistics, a driver is involved in an accident every six seconds. βπ₯βπ‘(π₯, π‘) Then we would rewrite the sentence to make it more accurate by using theexpression "every six seconds," instead of "second." In this case, we would say thatfor every six-second interval, there is a driver involved in an accident. βπ‘βπ₯π΄(π₯, π‘) As we can see here, the driver changes from one interval to another. For all valuesof t, there is an x such that X(t) = x, where these are differentiable functions. Theyare very different from each other. A lot of you had trouble seeing that there was anyproblem with the American Melanoma Foundation example, but if we make it preciseusing our formalism and language (language and formalism refer to mathematics orset theory), then we see that the order in which we state things makes a bigdifference in how we arrive at our conclusions. That was the whole point of thisexercise and these others; to make sure that left-to-right ordering of our formulascaptures their logical flow. And there is a logic to this. One sentence states that thereis a driver who's in an accident every six seconds, which makes no sense. Anothersentence states that for every six seconds, there's a driver in an accident. When welook at these statements formally, the distinction between them becomes clear.
Mathematical Thinking Intro: Equations, Primes, and More
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