Lecture 10 Everything Else 1 A Crash Course In Set Theory-Books Pdf

Lecture 10 Everything Else 1 A Crash Course in Set Theory
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The cartesian product of A and B denoted A B is the set of all ordered pairs. of the form a b that is a b a A b B, Sometimes we will have some larger set B like R out of which we will be picking. some subset A like 0 1 In this case we can form the complement of A with. respect to B namely Ac b B b A, With the concept of a set defined we can define functions as well. Definition A function f with domain A and codomain B formally speaking is a collec. tion of pairs a b with a A and b B such that there is exactly one pair a b for every. a A More informally a function f A B is just a map which takes each element in A. to some element of B,f Z N given by f n 2 n 1 is a function. g N N given by f n 2 n 1 is also a function It is in fact a different function. than f because it has a different domain, The function h depicted below by the three arrows is a function with domain 1. and codomain 24 Batman, This may seem like a silly example but it s illustrative of one key concept functions are.
just maps between sets Often people fall into the trap of assuming that functions. have to have some nice closed form like x3 sin x or something but that s not true. Often functions are either defined piecewise or have special cases or are generally fairly. ugly awful things in these cases the best way to think of them is just as a collection of. arrows from one set to another like we just did above. Functions have several convenient properties, Definition We call a function f injective if it never hits the same point twice i e for. every b B there is at most one a A such that f a b. Example The function h from before is not injective as it sends both and to 24. However if we add a new element to our codomain and make map to our function. is now injective as no two elements in the domain are sent to the same place. Definition We call a function f surjective if it hits every single point in its codomain. i e if for every b B there is at least one a A such that f a b. Alternately define the image of a function as the collection of all points that it maps. to That is for a function f A B define the image of f denoted f A as the set. b B a A such that f a b, Then a surjective function is any map whose image is equal to its codomain i e f. A B is surjective if and only if f A B, Example The function h from before is not injective as it doesn t send anything to. However if we add a new element to our domain and make map to Batman our. function is now surjective as it hits all of the elements in its codomain. Definition A function is called bijective if it is injective and surjective. Definition We say that two sets A B are the same size formally we say that they are of. the same cardinality and write A B if and only if there is a bijection f A B. Not all sets are the same size, Observation If A B are a pair of finite sets that contain different numbers of elements. If A is a finite set and B is infinite then A, If A is an infinite set such that there is a bijection A N call A countable If A is.
countable and B is a set that has a bijection to R then A. We can use this notion of size to make some more definitions. Definition We say that A B if and only if there is an injection f A B Similarly. we say that A B if and only if there is a surjection f A B. This motivates the following theorem, Theorem Cantor Schro der Bernstein Suppose that A B are two sets such that there. are injective functions f A B g B A Then A B i e there is some bijection. 2 A Crash Course in Combinatorics, Combinatorics very loosely speaking is the art of how to count things For the GRE a. handful of fairly simple techniques will come in handy. Multiplication principle Suppose that you have a set A each element a of which. can be broken up into n ordered pieces a1 an Suppose furthermore that the i th. piece has ki total possible states for each i and that our choices for the i th stage do. not interact with our choices for any other stage Then there are. k1 k2 kn ki,total elements in A,To give an example consider the following problem. Problem Suppose that we have n friends and k different kinds of postcards with. arbitrarily many postcards of each kind In how many ways can we mail out all of. our postcards to our friends, A valid way to mail postcards to friends is some way to assign each friend to a. postcard so that each friend is assigned to at least at least one postcard because. we re mailing each of our friends a postcard and no friend is assigned to two different. postcards at the same time In other words a way to mail postcards is just a. function from the set1 n 1 2 3 n of postcards to our set k 1 2 3 k. of friends, In other words we want to find the size of the following set.
A all of the functions that map n to k, We can do this Think about how any function f n k is constructed For each. value in n 1 2 n we have to pick exactly one value from k Doing this for. each value in n completely determines our function furthermore any two functions. f g are different if and only if there is some value m n at which we made a different. choice i e where f m 6 g m,k choices k choices k choices. n total slots,Consequently we have, total ways in which we can construct distinct functions This gives us this answer k n. to our problem, Summation principle Suppose that you have a set A that you can write as the. union2 of several smaller disjoint3 sets A1 An, Then the number of elements in A is just the summed number of elements in the Ai.
sets If we let S denote the number of elements in a set S then we can express this. in a formula,A A1 A2 An,We work one simple example. Some useful notation n denotes the collection of all integers from 1 to n i e 1 2 n. Given two sets A B we denote their union A B as the set containing all of the elements in either. A or B or both For example 2 lemur 2 lemur while 1 lemur 1 lemur. Sets are called disjointif they haven no elements in common For example 2 and lemur are disjoint. while 1 and lemur are not disjoint, Question 1 Pizzas Specifically suppose Pizza My Heart a local chain great pizza. place has the following deal on pizzas for 7 you can get a pizza with any two. different vegetable toppings or any one meat topping There are m meat choices and. v vegetable choices As well with any pizza you can pick one of c cheese choices. How many different kinds of pizza are covered by this sale. Using the summation principle we can break our pizzas into two types pizzas with. one meat topping or pizzas with two vegetable toppings. For the meat pizzas we have m c possible pizzas by the multiplication principle we. pick one of m meats and one of c cheeses, For the vegetable pizzas we have v2 c possible pizzas we pick two different vegetables. out of v vegetable choices and the order doesn t matter in which we choose them we. also choose one of c cheeses,Therefore in total we have c m v2 possible pizzas. Double counting principle Suppose that you have a set A and two different. expressions that count the number of elements in A Then those two expressions are. Again we work a simple example, Question 2 Without using induction prove the following equality.
First make a n 1 n 1 grid of dots, How many dots are in this grid On one hand the answer is easy to calculate it s. n 1 n 1 n2 2n 1, On the other hand suppose that we group dots by the following diagonal lines. The number of dots in the top left line is just one the number in the line directly. beneath that line is two the number directly beneath that line is three and so on so. forth until we get to the line containing the bottom left and top right corners which. contains n 1 dots From there as we keep moving right our lines go down by one in. size each time until we get to the line containing only the bottom right corner which. again has just one point, So if we use the summation principle we have that there are. 1 2 3 n 1 n n 1 n n 1 3 2 1,points in total, Therefore by our double counting principle we have just shown that. n2 2n 1 1 2 3 n 1 n n 1 n n 1 3 2 1, Rearranging the right hand side using summation notation lets us express this as.
n2 2n 1 n 1 2 i, subtracting n 1 from both sides and dividing by 2 gives us finally. which is our claim, Pigeonhole principle simple version Suppose that kn 1 pigeons are placed into. n pigeonholes Then some hole has at least k 1 pigeons in it In general replace. pigeons and pigeonholes with any collection of objects that you re placing in. various buckets,We look at an example, Question 3 Suppose that friendship is4 a symmetric relation i e that whenever. a person A is friends with a person B B is also friends with A Also suppose that. you are never friends with yourself5 i e that friendship is antireflexive. Just for this problem Be friends with yourself in real life. Then in any set S of greater than two people there are at least two people with the. same number of friends in S, Let S n Then every person in S has between 0 and n 1 friends in S Also notice. that we can never simultaneously have one person with 0 friends and one person with. n 1 friends at the same time because if someone has n 1 friends in S they must. be friends with everyone besides themselves, Therefore each person has at most n 1 possible numbers of friends and there are n.
people total by the pigeonhole principle if we think of people as the pigeons and. group them by their numbers of friends i e the pigeonholes are this grouping by. numbers of friends there must be some pair of people whose friendship numbers are. Some sorts of sets are very frequently counted, If we have a set of n objects there are n n n 1 1 many ways to order. this set For example the set a b c has 3 6 orderings. abc acb bac bca cab cba, Suppose that we have a set of n objects and we want to pick k of them without. repetition in order Then there are n n 1 n k 1 many ways to choose. them we have n choices for the first n 1 for the second and so on so forth until our. k th choice for which we have n k 1 choices We can alternately express this as. n k you can see this algebraically by dividing n by k or conceptually by thinking. our our choice process as actually ordering all n elements the n in our fraction and. then forgetting about the ordering on all of the elements after the first k as we didn t. pick them this divides by n k, Suppose that we have a set of n objects and we want to pick k of them without. repetition and without caring about the order in which we pick these k elements. Then there are k n k many ways for this to happen We denote this number as the. binomial coefficient nk, Finally suppose that we have a set of n objects and we want to pick k of them where. we can pick an element multiple times i e with repetition Then there are nk many. ways to do this by our multiplication principle from before. 3 A Crash Course in Probability,We give the basics of probability here.
Definition A finite probability space consists of two things. A finite set, A measure P r on such that P r 1 In case you haven t seen this before. saying that P r is a measure is a way of saying that P r is a function P R such. that the following properties are satisfied,For any collection Xi. i 1 of subsets of P r Xi P r Xi,For any collection Xi. i 1 of disjoint subsets of P r Xi P r Xi, For a general probability space i e one that may not be finite the definition is almost. completely the same the only difference is that is not restricted to be finite while P r. becomes a function defined only on the measurable subsets of For the GRE you can. probably assume that any set you run into is measurable There are some pathological. constructions in set theory that can be nonmeasurable talk to me to learn more about. For example one probability distribution on 1 2 3 4 5 6 could be the distri. bution that believes that P r i 1 6 for each individual i and more generally that. P r S S 6 for any subset S of In this sense this probability distribution is captur. ing the idea of rolling a fair six sided die and seeing what comes up. This sort of fair distribution has a name namely the uniform distribution. Definition The uniform distribution on a finite space is the probability space that. assigns the measure S to every subset S of In a sense this measure thinks that. any two elements in are equally likely think about why this is true. We have some useful notation and language for working with probability spaces. Definition An event S is just any subset of a probability space For example in the. six sided die probability distribution discussed earlier the set 2 4 6 is an event you can. think of this as the event where our die comes up as an even number The probability of. an event S occurring is just P r S i e the probability that our die when rolled is even is. just P r 2 4 6 3 6 1 2 as expected, Notice that by definition as P r is a measure for any two events A B we always have.
P r A B P r A P r B In other words given two events A B the probability of. either A or B happening or both is at most the probability that A happens plus the. probability that B happens, Definition A real valued random variable X on a probability space is simply any. function R, Given any random variable X we can talk about the expected value of X that is. the average value of X on where we use P r to give ourselves a good notion of what. average should mean Formally we define this as the following sum. For example consider our six sided die probability space again and the random variable X. defined by X i i in other words X is the random variable that outputs the top face of. the die when we roll it,The expected value of X would be. X 1 1 1 1 1 1 21 7,P r X 1 2 3 4 5 6,6 6 6 6 6 6 6 2. In other words rolling a fair six sided die once yields an average face value of 3 5. Definition Given a random variable X if E X the variance 2 X of X is just. E X 2 This can also be expressed as E X 2 E X 2 via some simple algebraic. manipulations, The standard deviation of X X is just the square root of the variance.
Definition Given a random variable X R on a probability space we can define. the density function for X denoted FX t as,FX t P r X t. Given this function we can define the probability density function fX t as dt Fx t. Notice that for any a b we have,P r a X b fX t dt, A random variable has a uniform distribution if its probability density function is a. constant this expresses the idea that uniformly distributed things don t care about the. differences between elements of our probability space. A random variable X with standard deviation expectation has a normal distri. bution if its probability density function has the form. fX t e t 2, This generates the standard bell curve picture that you ve seen in tons of different. settings One useful observation about normally distributed events is that about 68 of the. events occur within one standard deviation of the mean i e P r X 68. about 95 of events occur within two standard deviations of the mean and about 99 7. of events occur within three standard deviations of the mean. Definition For any two events A B that occur with nonzero probability define P r A. given B denoted P r A B as the likelihood that A happens given that B happens as well. Mathematically we define this as follows, In other words we are taking as our probability space all of the events for which B happens. and measuring how many of them also have A happen, Definition Take any two events A B that occur with nonzero probability We say that A.
and B are independent if knowledge about A is useless in determining knowledge about. B Mathematically we can express this as follows,P r A P r A B. Notice that this is equivalent to asking that,P r A P r B P r A B. Definition Take any n events A1 A2 An that each occur with nonzero probability We. say that these n events are are mutually independent if knowledge about any of these. Ai events is useless in determining knowledge about any other Aj Mathematically we can. express this as follows for any i1 ik and j 6 i1 ik we have. P r Aj P r Aj Ai1 Aik,It is not hard to prove the following results. Theorem A collection of n events A1 A2 An are mutually independent if and only if. for any distinct i1 ik 1 n we have,P r Ai1 Aik Aij. Theorem Given any event A in a probability space let Ac A denote. the complement of A, A collection of n events A1 A2 An are mutually independent if and only if their.
complements Ac1 Acn are mutually independent, It is also useful to note the following non result. Not theorem Pairwise independence does not imply independence In other words it is. possible for a collection of events A1 An to all be pairwise independent i e P r Ai. Aj P r Ai P r Aj for any i j but not mutually independent. Example There are many many examples One of the simplest is the following consider. the probability space generated by rolling two fair six sided dice where any pair i j of. faces comes up with probability 1 6,Consider the following three events. A the event that the first die comes up even,B the event that the second die comes up even. C the event that the sum of the two dice is odd, Each of these events clearly has probability 1 2 Moreover the probability of A B A C. and B C are all clearly 1 4 in the first case we are asking that both dice come up even. in the second we are asking for even odd and in the third asking for odd even all of. which happen 1 4 of the time So these events are pairwise independent as the probability. that any two happen is just the products of their individual probabilities. However A B C is impossible as A B holds iff the sum of our two dice is even. So P r A B C 0 6 P r A P r B P r C 1 8 and therefore we are not mutually.


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