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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 •AB •~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 : • 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 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 • 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 • 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 7 8 4 4 6 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