Math 55 - Fall 2007 - Lecture notes # 4 - Sep 5 (Wednesday) Keep Reading: Sections 2.1, 2.2, 2.3 Homework: Will be posted on web Today's goals: finish material on proofs from last lecture sets functions countability can one infinite set be bigger than another? are there programs to compute all possible functions? First goal: sets DEF: a set A is a collection of elements, also called members EG: A = {a,b,c}, A = {1,2,3,...,100}, A = { k^2 | k an integer, 1 <= k <= 100} = "the set of numbers k^2 such that k is an integer between 1 and 100" EG: Z = integers, N = nonnegative integers, Q = rational numbers, R = real numbers, R+ = nonnegative reals DEF: A = B as sets if same elements (order doesn't matter, multiple copies of elements does not matter, so {1,1,2} = {1,2} = {2,1}) DEF: NULLSET = {} = set with no elements DEF: x IN A is a proposition which is true if x is an element of A DEF: A SUBSET B means (x IN A) -> (x IN B) (Venn diagram) DEF: Intersection: "C = A inter B" means forall x, (x IN C) <-> (x IN A) and (x IN B) "C = A1 inter A2 inter ... An = inter_{i=1}^n Ai" means forall x, (x IN C) <-> ( (x IN A1) and (x IN A2) and ... and (x IN An) ) (Venn diagram) DEF: A inter B = NULLSET mean A and B are disjoint DEF: Union: "C = A union B" means forall x, (x IN C) <-> (x IN A) or (x IN B) "C = A1 union A2 union ... An = union_{i=1}^n Ai" means forall x, (x IN C) <-> ( (x IN A1) or (x IN A2) or ... or (x IN An) ) (Venn diagram) DEF: The Universal Set is the set of all possible elements a set can have EG: If U = "integers", A SUBSET U can be odd integers, perfect squares, etc. DEF: The complement of A, written bar(A) (A with a bar over it), is the set of all elements of U not in A Theorem (DeMorgan): bar(A union B) = bar(A) inter bar(B) Proof: (1) Venn Diagram (2) show that propositions x IN bar(A union B) and x IN bar(A) inter bar(B) are logically equivalent, using DeMorgan's law: x in bar(A union B) <=> not(x in A union B) <=> not (x in A or x in B) <=> (use DeMorgan for propositions) not(x in A) and not(x in B) <=> x in bar(A) and x in bar(B) <=> x in bar(A) inter bar(B) DEF: |A| = cardinality of A = number of elements in A (if A is finite, else say A is infinite, details of infinite case later) DEF: P(A) = power set of A = set of all subsets of A EG: A={1,2}, P(A)={NULLSET, {1}, {2}, {1,2}} ASK&WAIT: if |A| is finite, what is |P(A)|? DEF: (a1,...,an) is an ordered n-tuple ( parentheses() instead of braces{} means order matters) (a1,a2) is also called an ordered pair DEF: If A and B are sets, then A x B = { (a,b) | a in A and b in B } is the Cartesian product of A and B. A x B is also called the set of ordered pairs (a,b) from A, B EG: Suppose A and B are both set of real numbers. Then A x B is a 2-dimensional plane DEF: If A1, A2, ... , An are sets then A1 x A2 x ... x An = { (a1,a2,...,an) | ai in Ai for i=1,..,n } is the Cartesian product of A1,...,An. It is also called the set of ordered n-tuples (a1,...,an) from A1,..,An EG: A = {all keys on a keyboard, including return} = {a,b,...,z,A,B,...,Z,0,...,9,@,#,..,} A^2 = A x A = all ordered pairs of characters A^n = A x A x ... x A (n times) = all ordered n-tuples of characters S = A U A^2 U A^3 U ... = all finite strings of characters E = all syntactically correct English sentences E subset S J = all syntactically correct Java programs J subset S Second goal: functions DEF: Let A and B be sets. A function f from A to B (write f:A->B) is an assignment of exactly one element of B to each element of A (write f(a)=b to mean b IN B is assigned to a IN A). A is called the domain of A, and B is called the codomain. b=f(a) is the image of a, a is the preimage of b { f(a) | a IN A } is called the range of f (Figures 1, 2 in sec 2.3 show how functions may be represented) EG: f:Z->Z where f(z) = z^2, EG: f:Z->Z, where f(z) = 1, "constant function" EG: FLOOR:R->Z, where FLOOR(x) = largest integer <= x EG: CEILING:R->Z, where CEILING(x) = smallest integer >= x EG: LOG_2:R+->R, where R+ = nonnegative reals, LOG_2 = logarithm base 2 of x EG: f:{workers}->Z, where f(worker) = worker's Social Security # (SS#) (represented by table, not formula) EG: f:Z->{integer multiples of .01} where f(account number) = balance ASK&WAIT function f1(x), return x, end, function f2(x), return (2*x)/2, end What are domain and codomain? How can we choose the domain and codomain to make these functions equal? Are they the same functions on a computer? EG: Suppose f1:A->R and f2:A->R where A is any set, then f=f1+f2, g=f1*f2 etc are the functions satisfying f(x)=f1(x)+f2(x),g(x)=f1(x)*f2(x), etc ASK&WAIT: Let f1:R->R, where f1(x)=x, f2=f1, and f3:R->R, f3(x)=1. Does f1/f2 = f3 as functions? Generally, when is h=f1/f2 a function? How can we change h slightly to make it a function? What happens if we implement f1(x)/f2(x) on a computer? EG: B = {functions f_z:Z->Z | f_z(x)=x+z, z IN Z }, a set of functions g:Z->B, g(z) = f_z, i.e. function that adds z to its argument: g(z)(x) = f_z(x) = z+x DEF: f:A->B is one-to-one (injective) if f(x)=f(y) -> x=y ASK&WAIT is f:N->N where f(x) = x^2, injective? ASK&WAIT is f:Z->Z where f(x) = x^2, injective? ASK&WAIT is f:{workers}->Z where f(x) = SS#, injective?, DEF: f:A->B is onto (surjective) if all b IN B have preimages in A ASK&WAIT is f:Z->Z where f(x)=x+1 surjective? ASK&WAIT is f:N->N where f(x)=x+1 surjective? ASK&WAIT is f:{workers}->{9 decimal digit integers}, f(x) = SS#, surjective? ASK&WAIT: What does it mean if f:{drivers license numbers}->{names of actual drivers} where f(license number) = name of driver on license is not surjective? DEF: f:A->B is a one-to-one correspondence (bijective) if it is one-to-one and onto ASK&WAIT: is f:Z->Z where f(x)=x+1 bijective? ASK&WAIT: is f:Z->{even integers} where f(x)=2*x bijective? DEF: If f:A->B is a bijection, then the function f^{-1}:B->A defined by f{-1}(b)=a if f(a)=b is called the inverse function of f ASK&WAIT: if f:Z->Z, f(x)=x+1, what is f^{-1}? ASK&WAIT: if f:Z->{even integers}, f(x)=2*x, what is f^{-1}? ASK&WAIT: if f:{drivers license numbers}->{names of drivers}, what is f^{-1}? DEF: If g:A->B and f:B->C, then the function h:A->C defined by h(a)=f(g(a)) is called the composition of f and g, written h=f o g ASK&WAIT: f:R->R+, g:R+->R, (R+ = nonnegative reals) f(x)=x^2, g(x) = sqrt(x) What is f o g? (domain, codomain, range, value)? What is g o f? Is g o f = f o g ? DEF: id_A:A->A is the "identity function", if id_A(a)=a for all a IN A ASK&WAIT: let f:A->B be a bijection, and f^{-1}:B->A be inverse of f. What is f o f^{-1} ? What is f^{-1} o f? EG: f:{1,2,...,26}->{a,b,...,z} with f(1)=a,...,f(26)=z f o f^{-1} is identity on {a,b,...,z} f^{-1} o f is identity on {1,2,...,26} DEF: If f:A->B, then the graph of f is the set of ordered pairs { (a,b) | a IN A and f(a)=b } ASK&WAIT: what is graph of f:R->R, f(x)=x^2? ASK&WAIT: what is graph of f:{workers}->Z, f(worker)=SSN? Third Goal: understand cardinality, countability Recall DEF: If A is finite, the cardinality |A| = # members of A ASK&WAIT: suppose f:A->B is a bijection, A finite. Is B finite? How are |A| and |B| related? DEF: We say that A and B have the same cardinality if there is a one-to-one correpondence between them, whether finite or not ASK&WAIT: Do Z and {even integers} have same cardinality? ASK&WAIT: Do N and {powers of 2} have same cardinality? ASK&WAIT: Do Z and N have same cardinality? (hint: represent bijection by table) DEF: A set that is either finite or has the same cardinality as N (or Z) is called countable, else uncountable intuition is that an uncountable set is much larger than any countable set More examples of countable sets (most sets we have seen are countable:) Theorem: if A and B are countable, so is S = A union B proof: number elements of S by a(1), b(1), a(2), b(2),... i.e. f(i) = a(i/2) if i is even; b((i+1)/2) if i is odd is a one-to-one correspondence between N and S Enough to illustrate bijection f:N->S of a set S with N or Z without writing down formula for f, i.e. just show how to write down all members of S in order each member of S appearing exactly once Theorem: The Cartesian product P = A x B of all pairs {(a,b)} is countable if A and B are countable proof: represent P as "lattice" points in the plane, and number them diagonally. Theorem: Suppose A1, A2, A3, ... are all infinite countable sets Then S = A1 union A2 union A3 union ... is countable ASK&WAIT: why? Theorem: Suppose A is countable, and B is a subset of A. Then B is countable ASK&WAIT: why? ASK&WAIT: Is Q (rational numbers) countable? ASK&WAIT: Is J (set of syntactically correct programs) countable?