Note for MATH 410 with Professor Martin at KU
Note for MATH 410 with Professor Martin at KU
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Date Created: 02/06/15
In nity 1 Problems involving in nity What does in nite mean How do you measure the size of an in nite set Do line segments of different lengths have the same number of points Are there more points in a big line segment or a small triangle These questions are hard and have occupied mathematicians for millennia To make things a bit more concrete consider two line segments of lengths l and 2 for example the intervals A 01 and B 02 on the number line Strictly speaking we re thinking of each ofA and B as a set of real numbers For example 7r4 E A 7r2 E B and 7r2 g A1 On the one hand A is a proper subset of B that is every point in A is also in B but not vice versa so it is plausible that A should have fewer points than B does On the other hand both sets are clearly in nite so they ought to have the same size Actually neither of these arguments is correct Just because A is a subset of B does Lot mean that A has fewer points than B In fact these two intervals have exactly the same sizes On the other hand something even more mindboggling is true there are lots of different sizes of in nite sets 7 in fact in nitely many different in nitiesl 2 Greek ideas about in nity The ancient Greeks regarded in nity not as a number but as sort of an unattainable concept Aristotle distinguished between actual in nity which doesnlt exist and potential in nity which does An example of this is Euclidls prime number theorem see Prime numbers from the course notes Today we would phrase the theorem as The set of all prime numbers is in nite but Euclid said No nite set of prime numbers can possibly be the complete list of all prime numbers Another kind of phrasing you might see in ancient mathematics No nite set of prime numbers exhausts all the prime numbers The Greeks didn t mind this they found ways to do mathematics without resorting to in nity An example is Archimedes7 quadrature of the parabola whose ideas anticipate modern integral calculus but are phrased differently 1Remember the symbols 6 and mean is a member of and is not a member of respectively A parabolic segment is the region bounded by a piece of a parabola cut off by a line segment such as the red shaded region P shown above Archimedes wanted to calculate the area of a parabolic segmenti Today we7d probably use calculus to express the area as a de nite integral but Archimedes didn t have calculus in his toolkit for that matter he didnlt even have coordinatesi Instead he came up with the following geometric construction 1 Find the point C on the parabola such that the tangent line at C is parallel to El Draw the triangle AABCi See gure above center 2 Find points DE on the parabola where the tangent lines are parallel to AC and BC respectivelyi Draw the triangles AADC and ACEBi See gure above right 3 Repeat this process to draw four more triangles say AAWD ADXC ACYE AEZB not shown 4 Repeat this process to draw eight more trianglesi Next Archimedes showed that each triangle has oneeighth the area of its parent trianglei Let a be the area of AABC then areaAADC areaACEB 18 areaAAWD areaADXC areaACYE areaAEZB 164 etc So the total area covered in the rst n steps is 1 l i L a 416quot394n71 This is a partial sum of a geometric series it is equal to lt1 G W A modern mathematician would say that the area of the parabolic segment is therefore 4a 1 4a JEEOElt1 lt1gtgt Archimedes remember didn t have tools like limitsi Instead he reasoned as follows 0 Any number I lt 4a3 must be less than the area of P because if you repeat this process enough times the area covered by the triangles eventually exceeds 1 0 Any number y gt 4a3 must be greater than the actual area of P because no matter how many triangles you draw the area they cover namely the quantity in formula is always less than 4a3i 0 Therefore the area of P must be exactly 4a3i This reasoning is perfectly correct and actually contains some fairly deep ideas about limits and sequences and things hidden inside it 7 but notice how it s phrased so as to avoid any explicit mention of in nity or limits or convergence or in nitesimalsi 3 Bijections Suppose that F and G are two sets How can we tell if F and G are the same size For that matter what does size or cardinality which is the technical term mean To say that F has n elements symbolically F n is to say that there s a function qugt 12iiin that is onetoone and onto Such a function is called a bijection So F G if there exist bijections qFgtl2iHn TGgtl2uin for some nonnegative integer n On the other hand we don7t need to know the actual size of two sets to know that they are the same cardinality We can just say that F G if there exists a bijection b F A G 7 in other words if it is possible to pair the elements of F with the elements of C so that every element of F has exactly one mate in Cr This de nition of same size77 has the advantage that it applies to in nite sets as well A good way to think about a bijection between two sets is as a way of labelling the elements of one set with the elements of the other set using eacj label exactly oncei Under this de nition the intervals 01 and 0 2 have the same cardinality because there is a bijection between them namely f 01 A 02 de ned by 21 It doesn7t matter that the rst interval is a proper subset of second 7 in fact the rule If A is a proper subset of B then A lt B77 applies only if A is nite This de nition while sensible has a number of striking consequencesi For example the function 21 de nes a bijection between the in nite sets N0123m 1E0246mi So N lE even though there are in nitely many numbers that are in N but not in E In addition the sets Z2 ordered pairs of integers N2 ordered pairs of natural numbers and even Q rational numbers have the same cardinality as N that is they are countably in nitei For example to label all the elements of N2 with the elements of N we can arrange the elements of N2 in a square grid draw a zig zag path that covers all of them then label the path in order 04 10 03 13 9 3 02 8 12 22 2 01 4 11 7 21 31 00 10 20 gt 30 40 0 1 5 6 On the other hand the set R of real numbers is Lot countably in nite The astonishing truth proved by Georg Cantor is that there can exist no bijection between N and R Stay tuned
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