Mathematical Preliminaries Sets, Set Operations, and Boolean Algebra Logic and Proof Techniques Set Product, Relations and Functions Equivalence Relations and Partial Orders Countable Sets Recursive Definitions Graphs

Sets, Set Operations Sets : A set is an unordered collection of objects, usually with some common property Examples – set of integers, set of symbols {alphabet}, set of syntactic types, set of symbol strings (language) Subsets : A is a subset of B (written A  B) if every element of A is also an element of B (written  x  A  x  B) Exercise : Show transitivity of  (i.e. A  B and B  C  A  C ) Set Operations :  : x  A  B iff x  A or x  B  : x  A  B iff x  A and x  B ~ : x  ~A iff x  A Characteristic Vectors : If A is a set of n elements (cardinality n), and B  A, then Bv is a binary vector with ith component corresponding to ith element xi of A and vi = 1 if xi  A and vi = 0 otherwise. Example: A = {a,b,c,d}. Then 0101 represents B = {b,d} and 1001 represents C = {a,d}. Exercise : Describe the construction of the characteristic vector of A  B A  B ~ A

Set Specification Set specification : A set is specified by giving a rule or definition which determines which objects are in the set. List – if the set is finite, a list of objects belonging to the set is often used to specify the set. Exercise : Write pseudo code to perform set operations for two sets specified by ordered lists. For infinite or large finite sets, the following methods of specification are used: Property – A property possesed by all and only those set elements is given. Acceptor – A finite state acceptor is used for languages (sets of strings) for which only a finite number of things need to be remembered. Recursive methods – a finite basis set is given along with rules for forming the reset of the elements from existing elements. Grammars – Languages are specified by a finite set of rules which either give basis elements or tell how to build more complex strings from simpler ones.

Boolean Algebra Boolean Algebra : Subset Specification Two binary operations ( + , * ) and a unary operation ( ~ ) defined as follows: 0 + 0 = 0, 0 + 1 = 1 + 0 = 1, 1 + 1 = 1 0 * 0 = 0 * 1 = 1 * 0 = 0, 1 * 1 = 1 ~0 = 1 and ~1 = 0 Subset Specification Let U be a set {x1, x2, . . Xn} and A any subset. Subset A can be specified by a boolean vector of n bits called the characteristic vector of A The ith bit of the characteristic vector = 1 if xi is in A and 0 otherwise. Exercise : Describe the relation between Boolean Algebra and Set Operations. Exercise : Write pseudo code to perform Union, Intersection and Complement given charateristic vectors.

Propositional Logic A declarative statement such as “Bill is a CS student” has a truth value of T or F and is denoted by P (a truth variable) Propositions may be combined with logical operators and the composite statement has value as shown below. P  Q is true if either P or Q are true and false if both are false P  Q is true if both P and Q are true and false if either is false. ¬ P is true if P is false and false if P is true P  Q is true if P and Q have the same truth value and false if their values differ P  Q is false if P is true and Q is false and true otherwise. Exercise – Construct truth tables for each operation. A tautology is always true. P  Q  ¬ P  Q is a tautology. P  (Q  R)  (P  Q)  (P  R) is a tautology.

Rules of Inference P , P  Q then Q - modus ponens ¬ Q, P  Q then ¬ P - modus tollens Exercise : Show by truth table. Induction P1 is true Pi  Pi+1 is true for all i Then Pi is true for all i

Pi : Sum of integers from 1 to n is n(n+1)/2. Example of Induction Pi : Sum of integers from 1 to n is n(n+1)/2. P1 : Sum of integers from 1 to 1 is 1 which equals 1(1+1)/2 so P1 true. Pi  Pi+1 : If Pi is true, then sum of integers from 1 to n+1 is sum of integers from 1 to n + n+1, which is n(n+1)/2 + (n+1) = (n/2 + 1)(n+1) = (n+2)(n+1)/2 so Pi+1 is true. so Pi  Pi+1 is true for all i

Cartesian Set Product and Relations Cartesian Product The product of two sets A = {a1, .. am} and B = {b1, .. bn} is a set, denoted by A  B, of ordered pairs (ai,bj) of cardinality m*n. Example : A = {a,b} and B = {1,2,3} A  B = {(a,1),(a,2),(a,3),(b,1),(b,2),(b,3)} Relations A relation R between two sets A,B is a subset of their product A  B. That is, R is a relation between A,B if R  A  B Example :  is a relation between {1,2,3} and itself since  is a subset of {1,2,3}  {1.2.3} Exercise : What are the elements of  as a relation between {1,2,3} and itself?

Functions  a  A,  b  B for which (a,b)  f written f(a) = b Functions : A function f from A into B. written f : A  B where A is called the domain of f and B the range, is a relation between A and B for which  a  A,  b  B for which (a,b)  f written f(a) = b for each input in the domain A, there is an output if (a,b1)  f and (a,b2)  f, then b1 = b2. the output is unique. Exercise : Consider A = {1,2,3} and B = {a,b} Which of the following are functions from A into B? R = {(2,b),(3,a)} S = {(3,b),(2,a),(1,a)} T = {(1,b),(2,a),(1,a)}

Properties of Functions Onto Functions A function f : D → R is onto the range R if every element of R occurs as an output for some input in the domain D. f : {1,2} → {a,b,c} with f(1) = a, f(2) = c is not onto {a,b,c} because b does not occur as an output for any input. g : {1,2,3} → {a,b} with f(1) = a, f(2) = a and f(3) = b is onto {a,b} because both a and b occur as outputs for some input 1-1 Functions A function f : D → R is 1-1 if every element of R occurs as at most one output of some input in the domain D. f : {1,2} → {a,b,c} with f(1) = a, f(2) = c is a 1-1 function. g : {1,2,3} → {a,b} with f(1) = a, f(2) = a and f(3) = b is not 1-1. Inverse function The inverse f -1 of a function f maps the range onto the domain as follows: f -1(b) = a iff f(a) =b.

Power Sets If A is a set, its power set 2A is the set of all subsets of A. Exercise: Construct the power set of A = {a,b,c} Exercise: Construct the function f which has 2A as domain and the set of corresponding characteristic vectors as range.

Equivalence Relations and Partial Orders Let R be a relation on A so R  R  R R is reflexive if a R a a, a  A R is symmetric if a R b  b R a R is anti-symmetric if a R b and b R a  a=b R is transitive if a R c if a R b and b R c. A relation which is reflexive, symmetric and transitive is called an equivalence relation. An equivalence relation partitions a set A into disjoint equivalence classes. A relation which is reflexive, anti-symmetric and transitive is called a partial order.

Countable Sets Finite Sets If  a 1-1, onto function from A onto {1,..,n} then A is finite of cardinality n. If A is finite of cardinality n then  a 1-1, onto function from A onto {1,..,n}. Infinite sets A is countable (and infinite) if  a 1-1, onto function from A onto the postive integers. Exercise : Show the set of integers – {0} is countable

Exercise : Show the set of integers – {0} is countable Construct a 1-1 onto function F from set of integers – {0} onto set of positive integers Set of integers – {0} = {negative integers}  {positive integers} Let x be negative integer : F(x) = 2 * |x+1| + 1 so F(-1) = 1, F(-2) = 3, F(-3) = 5, .. So sub range of F for negative integers is odd integers Let x be positive integer : F(x) = 2*x so F(1) = 2, F(2) = 4, F(3) = 6, .. So sub range of F for positive integers is even integers

Recursive Definitions Peano’s Axioms (for the natural numbers) A recursive definition of N, the set of natural numbers, is constructed using the success function s : s(n) = n+1 Basis : 0 € N Recursive step: If n € N, then s(n) € N Closure : n € N   p € N and s(p) = n

Graphs the elements of V called the vertices of G A graph G consists of a relation E on a set V with the elements of V called the vertices of G the elements of E called the edges of G the graph is directed if the relation is not symmetric and undirected if the relation is symmetric. Let E be the relation “is a factor of” on {1,2,3,4,5,6,7,8}. A diagram of the graph of this relation is show below. 2 1 3 5 6 44 8 7

Paths, Circuits and Trees Paths : A simple path in a graph is a sequence of vertices v1, v2 . . Vn such that for i = 1 to n-1, (vi,vi+1) is an edge of the graph and vertices are distinct, except for possible the first and last. Connected : A vertex u is connected to every vertex v for which there is a path from u to v. Connected Graph : A connected graph is one in which every vertex is connected to every other vertex. Circuits : A circuit is a simple path for which the first and last vertices are the same. Tree : A tree is a graph with no circuits is connected. Exercise. Prove that a tree with n vertices has n-1 edges.

A tree with n vertices has n-1 edges. Basis : A tree with 1 vertex has 0 edges. Inductive Step: If a tree with n vertices has n-1 edges, show that a tree with n+1 vertices has n edges. Let T be a tree with n+1 vertices. Find a vertex v of degree 1 so edge e = (v,u) is only edge incident with v. Delete v and e so remainder is tree (why?) with n vertices and must have n-1 edges by inductive hypothesis. Original tree has (n-1) + 1 = n edges. QED