Math 94 Professor: Padraic Bartlett Lecture 10: Everything Else Week 10 UCSB 2015 This is the tenth week of the Mathematics Subject Test GRE prep course; here, we quickly review a handful of useful concepts from the fields of probability, combinatorics, and set theory! As always, each of these fields are things you could spend years studying; we present here a few very small slices of each topic that are particularly key to each. 1 A Crash Course in Set Theory Sets are, in a sense, what mathematics are founded on. For the GRE, much of the intricate parts of set theory (the axioms of set theory, the axiom of choice, etc.) aren't particularly relevant; instead, most of what we need to review is simply the language and notation for set theory. Definition. A set S is simply any collection of elements. We denote a set S by its elements, which we enclose in a set of parentheses. For example, the set of female characters in Measure for Measure is fIsabella, Mariana, Juliet, Franciscag: Another way to describe a set is not by listing its elements, but by listing its properties. For example, the even integers greater than π can be described as follows: fx j π < x; x 2 Z; 2 divides xg: Definition. If we have two sets A; B, we write A ⊆ B if every element of A is an element of B. For example, Z ⊆ R. Definition. A specific set that we often care about is the empty set, ; i.e. the set con- taining no elements. One particular quirk of the empty set is that any statement of the form 8x 2 ; ::: will always be vacuously true, as it is impossible to disprove (as we disprove 8 claims by using 9 quantifiers!) For example, the statement \every element of the empty set is delicious" is true. Dumb, but true! Some other frequently-occurring sets are the open intervals (a; b) = fx j x 2 R; a < x < bg and closed intervals [a; b] = fx j x 2 R; a ≤ x ≤ bg. Definition. Given two sets A; B, we can form several other particularly useful sets: • The difference of A and B, denoted A−B or AnB. This is the set fx j x 2 A; x2 = Bg. • The intersection of A and B, denoted A \ B, is the set fx j x 2 A and x 2 Bg. • The union of A and B, denoted A[B, is the set fx j x 2 A or x 2 B; or possibly both.g. 1 • The cartesian product of A and B, denoted A × B, is the set of all ordered pairs of the form (a; b); that is, f(a; b) j a 2 A; b 2 Bg. • 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 = fb 2 B j b2 = Ag. 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 2 A and b 2 B; such that there is exactly one pair (a; b) for every a 2 A. More informally, a function f : A ! B is just a map which takes each element in A to some element of B. Examples. • f : Z ! N given by f(n) = 2jnj + 1 is a function. • g : N ! N given by f(n) = 2jnj + 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 f1; λ, 'g and codomain f24; γ; Batmang : 1 24 ; ? λ # γ ' 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 2 B, there is at most one a 2 A such that f(a) = b. Example. The function h from before is not injective, as it sends both λ and ' to 24: 2 1 24 ; ? λ # γ ' Batman However, if we add a new element π to our codomain, and make ' map to π, our function is now injective, as no two elements in the domain are sent to the same place: 1 24 ; λ # γ ' Batman * π Definition. We call a function f surjective if it hits every single point in its codomain { i.e. if for every b 2 B, there is at least one a 2 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 fb 2 B j 9a 2 A such that f(a) = bg. 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 Batman: 1 24 ; ? λ # γ ' Batman However, if we add a new element ρ to our domain, and make ρ map to Batman, our function is now surjective, as it hits all of the elements in its codomain: 3 1 24 ; ? λ # γ ' Batman C ρ 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 jAj = jBj, 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, then jAj 6= jBj. If A is a finite set and B is infinite, then jAj= 6 jBj. 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 jAj= 6 jBj. We can use this notion of size to make some more definitions: Definition. We say that jAj ≤ jBj if and only if there is an injection f : A ! B. Similarly, we say that jAj ≥ jBj if and only if there is a surjection f : A ! B. This motivates the following theorem: Theorem. (Cantor-Schr¨oder-Bernstein):Suppose that A; B are two sets such that there are injective functions f : A ! B; g : B ! A. Then jAj = jBj; i.e. there is some bijection h : A ! B. 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 n Y k1 · k2 · ::: · kn = ki i=1 total elements in A. To give an example, consider the following problem: 4 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] = f1; 2; 3; : : : ng of postcards to our set [k] = f1; 2; 3; : : : kg of friends! In other words, we want to find the size of the following set: n o 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] = f1; 2; : : : ng, 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 2 [n] at which we made a different choice (i.e. where f(m) 6= g(m).) k choices · k choices · ::: · k choices | {z } n total slots Consequently, we have k · k · ::: · k = kn | {z } n total ways in which we can construct distinct functions.